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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5% ================================================================
6% Chapter 1  Model Basics
7% ================================================================
8\chapter{Model Basics}
9\label{chap:PE}
10\minitoc
11
12\newpage
13
14% ================================================================
15% Primitive Equations
16% ================================================================
17\section{Primitive equations}
18\label{sec:PE_PE}
19
20% -------------------------------------------------------------------------------------------------------------
21%        Vector Invariant Formulation
22% -------------------------------------------------------------------------------------------------------------
23
24\subsection{Vector invariant formulation}
25\label{subsec:PE_Vector}
26
27The ocean is a fluid that can be described to a good approximation by the primitive equations,
28\ie the Navier-Stokes equations along with a nonlinear equation of state which
29couples the two active tracers (temperature and salinity) to the fluid velocity,
30plus the following additional assumptions made from scale considerations:
31
32\begin{enumerate}
33\item
34  \textit{spherical earth approximation}: the geopotential surfaces are assumed to be spheres so that
35  gravity (local vertical) is parallel to the earth's radius
36\item
37  \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius
38\item
39  \textit{turbulent closure hypothesis}: the turbulent fluxes
40  (which represent the effect of small scale processes on the large-scale)
41  are expressed in terms of large-scale features
42\item
43  \textit{Boussinesq hypothesis}: density variations are neglected except in their contribution to
44  the buoyancy force
45  \begin{equation}
46    \label{eq:PE_eos}
47    \rho = \rho \ (T,S,p)
48  \end{equation}
49\item
50  \textit{Hydrostatic hypothesis}: the vertical momentum equation is reduced to a balance between
51  the vertical pressure gradient and the buoyancy force
52  (this removes convective processes from the initial Navier-Stokes equations and so
53  convective processes must be parameterized instead)
54  \begin{equation}
55    \label{eq:PE_hydrostatic}
56    \pd[p]{z} = - \rho \ g
57  \end{equation}
58\item
59  \textit{Incompressibility hypothesis}: the three dimensional divergence of the velocity vector $\vect U$
60  is assumed to be zero.
61  \begin{equation}
62    \label{eq:PE_continuity}
63    \nabla \cdot \vect U = 0
64  \end{equation}
65\end{enumerate}
66
67Because the gravitational force is so dominant in the equations of large-scale motions,
68it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the earth such that
69$k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$,
70\ie tangent to the geopotential surfaces.
71Let us define the following variables: $\vect U$ the vector velocity, $\vect U = \vect U_h + w \, \vect k$
72(the subscript $h$ denotes the local horizontal vector, \ie over the $(i,j)$ plane),
73$T$ the potential temperature, $S$ the salinity, $\rho$ the \textit{in situ} density.
74The vector invariant form of the primitive equations in the $(i,j,k)$ vector system provides
75the following equations:
76\begin{subequations}
77  \label{eq:PE}
78  \begin{gather}
79    \intertext{$-$ the momentum balance}
80    \label{eq:PE_dyn}
81    \pd[\vect U_h]{t} = - \lt[ (\nabla \times \vect U) \times \vect U + \frac{1}{2} \nabla \lt( \vect U^2 \rt) \rt]_h
82                        - f \; k \times \vect U_h - \frac{1}{\rho_o} \nabla_h p
83                        + \vect D^{\vect U} + \vect F^{\vect U} \\
84    \intertext{$-$ the heat and salt conservation equations}
85    \label{eq:PE_tra_T}
86    \pd[T]{t} = - \nabla \cdot (T \ \vect U) + D^T + F^T \\
87    \label{eq:PE_tra_S}
88    \pd[S]{t} = - \nabla \cdot (S \ \vect U) + D^S + F^S
89  \end{gather}
90\end{subequations}
91where $\nabla$ is the generalised derivative vector operator in $(i,j,k)$ directions, $t$ is the time,
92$z$ is the vertical coordinate, $\rho$ is the \textit{in situ} density given by the equation of state
93(\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,
94$f = 2 \vect \Omega \cdot k$ is the Coriolis acceleration
95(where $\vect \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration.
96$\vect D^{\vect U}$, $D^T$ and $D^S$ are the parameterisations of small-scale physics for momentum,
97temperature and salinity, and $\vect F^{\vect U}$, $F^T$ and $F^S$ surface forcing terms.
98Their nature and formulation are discussed in \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}.
99
100% -------------------------------------------------------------------------------------------------------------
101% Boundary condition
102% -------------------------------------------------------------------------------------------------------------
103\subsection{Boundary conditions}
104\label{subsec:PE_boundary_condition}
105
106An ocean is bounded by complex coastlines, bottom topography at its base and
107an air-sea or ice-sea interface at its top.
108These boundaries can be defined by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,k,t)$,
109where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface.
110Both $H$ and $\eta$ are usually referenced to a given surface, $z = 0$, chosen as a mean sea surface
111(\autoref{fig:ocean_bc}).
112Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with
113the solid earth, the continental margins, the sea ice and the atmosphere.
114However, some of these fluxes are so weak that even on climatic time scales of thousands of years
115they can be neglected.
116In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and
117the other components of the earth system.
118
119%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
120\begin{figure}[!ht]
121  \begin{center}
122    \includegraphics[]{Fig_I_ocean_bc}
123    \caption{
124      \protect\label{fig:ocean_bc}
125      The ocean is bounded by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,t)$,
126      where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface.
127      Both $H$ and $\eta$ are referenced to $z = 0$.
128    }
129  \end{center}
130\end{figure}
131%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
132
133\begin{description}
134\item[Land - ocean interface:]
135  the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff.
136  Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths.
137  It can be neglected for short range integrations but has to be taken into account for long term integrations as
138  it influences the characteristics of water masses formed (especially at high latitudes).
139  It is required in order to close the water cycle of the climate system.
140  It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths.
141\item[Solid earth - ocean interface:]
142  heat and salt fluxes through the sea floor are small, except in special areas of little extent.
143  They are usually neglected in the model
144  \footnote{
145    In fact, it has been shown that the heat flux associated with the solid Earth cooling
146    (\ie the geothermal heating) is not negligible for the thermohaline circulation of the world ocean
147    (see \autoref{subsec:TRA_bbc}).
148  }.
149  The boundary condition is thus set to no flux of heat and salt across solid boundaries.
150  For momentum, the situation is different. There is no flow across solid boundaries,
151  \ie the velocity normal to the ocean bottom and coastlines is zero (in other words,
152  the bottom velocity is parallel to solid boundaries). This kinematic boundary condition
153  can be expressed as:
154  \begin{equation}
155    \label{eq:PE_w_bbc}
156    w = - \vect U_h \cdot \nabla_h (H)
157  \end{equation}
158  In addition, the ocean exchanges momentum with the earth through frictional processes.
159  Such momentum transfer occurs at small scales in a boundary layer.
160  It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions.
161  Its specification depends on the nature of the physical parameterisation used for
162  $\vect D^{\vect U}$ in \autoref{eq:PE_dyn}.
163  It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9.
164\item[Atmosphere - ocean interface:]
165  the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget)
166  leads to:
167  \[
168    % \label{eq:PE_w_sbc}
169    w = \pd[\eta]{t} + \lt. \vect U_h \rt|_{z = \eta} \cdot \nabla_h (\eta) + P - E
170  \]
171  The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system)
172  leads to the continuity of pressure across the interface $z = \eta$.
173  The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
174\item[Sea ice - ocean interface:]
175  the ocean and sea ice exchange heat, salt, fresh water and momentum.
176  The sea surface temperature is constrained to be at the freezing point at the interface.
177  Sea ice salinity is very low ($\sim4-6 \, psu$) compared to those of the ocean ($\sim34 \, psu$).
178  The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected.
179\end{description}
180
181% ================================================================
182% The Horizontal Pressure Gradient
183% ================================================================
184\section{Horizontal pressure gradient}
185\label{sec:PE_hor_pg}
186
187% -------------------------------------------------------------------------------------------------------------
188% Pressure Formulation
189% -------------------------------------------------------------------------------------------------------------
190\subsection{Pressure formulation}
191\label{subsec:PE_p_formulation}
192
193The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at
194a reference geopotential surface ($z = 0$) and a hydrostatic pressure $p_h$ such that:
195$p(i,j,k,t) = p_s(i,j,t) + p_h(i,j,k,t)$.
196The latter is computed by integrating (\autoref{eq:PE_hydrostatic}),
197assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}).
198The hydrostatic pressure is then given by:
199\[
200  % \label{eq:PE_pressure}
201  p_h (i,j,z,t) = \int_{\varsigma = z}^{\varsigma = 0} g \; \rho (T,S,\varsigma) \; d \varsigma
202\]
203Two strategies can be considered for the surface pressure term:
204$(a)$ introduce of a  new variable $\eta$, the free-surface elevation,
205for which a prognostic equation can be established and solved;
206$(b)$ assume that the ocean surface is a rigid lid,
207on which the pressure (or its horizontal gradient) can be diagnosed.
208When the former strategy is used, one solution of the free-surface elevation consists of
209the excitation of external gravity waves.
210The flow is barotropic and the surface moves up and down with gravity as the restoring force.
211The phase speed of such waves is high (some hundreds of metres per second) so that
212the time step would have to be very short if they were present in the model.
213The latter strategy filters out these waves since the rigid lid approximation implies $\eta = 0$,
214\ie the sea surface is the surface $z = 0$.
215This well known approximation increases the surface wave speed to infinity and
216modifies certain other longwave dynamics (\eg barotropic Rossby or planetary waves).
217The rigid-lid hypothesis is an obsolescent feature in modern OGCMs.
218It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings.
219Only the free surface formulation is now described in the this document (see the next sub-section).
220
221% -------------------------------------------------------------------------------------------------------------
222% Free Surface Formulation
223% -------------------------------------------------------------------------------------------------------------
224\subsection{Free surface formulation}
225\label{subsec:PE_free_surface}
226
227In the free surface formulation, a variable $\eta$, the sea-surface height,
228is introduced which describes the shape of the air-sea interface.
229This variable is solution of a prognostic equation which is established by forming the vertical average of
230the kinematic surface condition (\autoref{eq:PE_w_bbc}):
231\begin{equation}
232  \label{eq:PE_ssh}
233  \pd[\eta]{t} = - D + P - E \quad \text{where} \quad D = \nabla \cdot \lt[ (H + \eta) \; \overline{U}_h \, \rt]
234\end{equation}
235and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
236
237Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as
238a class of solution of the primitive equations.
239These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high.
240Their time scale is short with respect to the other processes described by the primitive equations.
241
242Two choices can be made regarding the implementation of the free surface in the model,
243depending on the physical processes of interest.
244
245$\bullet$ If one is interested in EGWs, in particular the tides and their interaction with
246the baroclinic structure of the ocean (internal waves) possibly in shallow seas,
247then a non linear free surface is the most appropriate.
248This means that no approximation is made in \autoref{eq:PE_ssh} and that
249the variation of the ocean volume is fully taken into account.
250Note that in order to study the fast time scales associated with EGWs it is necessary to
251minimize time filtering effects
252(use an explicit time scheme with very small time step, or a split-explicit scheme with reasonably small time step,
253see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}).
254
255$\bullet$ If one is not interested in EGW but rather sees them as high frequency noise,
256it is possible to apply an explicit filter to slow down the fastest waves while
257not altering the slow barotropic Rossby waves.
258If further, an approximative conservation of heat and salt contents is sufficient for the problem solved,
259then it is sufficient to solve a linearized version of \autoref{eq:PE_ssh},
260which still allows to take into account freshwater fluxes applied at the ocean surface \citep{roullet.madec_JGR00}.
261Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost.
262
263The filtering of EGWs in models with a free surface is usually a matter of discretisation of
264the temporal derivatives,
265using a split-explicit method \citep{killworth.webb.ea_JPO91, zhang.endoh_JGR92} or
266the implicit scheme \citep{dukowicz.smith_JGR94} or
267the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}.
268With the present release, \NEMO offers the choice between
269an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or
270a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05}
271(see \autoref{subsec:DYN_spg_ts}).
272
273% ================================================================
274% Curvilinear z-coordinate System
275% ================================================================
276\section{Curvilinear \textit{z-}coordinate system}
277\label{sec:PE_zco}
278
279% -------------------------------------------------------------------------------------------------------------
280% Tensorial Formalism
281% -------------------------------------------------------------------------------------------------------------
282\subsection{Tensorial formalism}
283\label{subsec:PE_tensorial}
284
285In many ocean circulation problems, the flow field has regions of enhanced dynamics
286(\ie surface layers, western boundary currents, equatorial currents, or ocean fronts).
287The representation of such dynamical processes can be improved by
288specifically increasing the model resolution in these regions.
289As well, it may be convenient to use a lateral boundary-following coordinate system to
290better represent coastal dynamics.
291Moreover, the common geographical coordinate system has a singular point at the North Pole that
292cannot be easily treated in a global model without filtering.
293A solution consists of introducing an appropriate coordinate transformation that
294shifts the singular point onto land \citep{madec.imbard_CD96, murray_JCP96}.
295As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems.
296An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism.
297This formalism is suited to any multidimensional curvilinear coordinate system.
298Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation),
299with preservation of the local vertical. Here we give the simplified equations for this particular case.
300The general case is detailed by \citet{eiseman.stone_SR80} in their survey of the conservation laws of fluid dynamics.
301
302Let $(i,j,k)$ be a set of orthogonal curvilinear coordinates on
303the sphere associated with the positively oriented orthogonal set of unit vectors
304$(i,j,k)$ linked to the earth such that
305$k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$,
306\ie along geopotential surfaces (\autoref{fig:referential}).
307Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by
308the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and
309the distance from the centre of the earth $a + z(k)$ where $a$ is the earth's radius and
310$z$ the altitude above a reference sea level (\autoref{fig:referential}).
311The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$,
312the three scale factors:
313\begin{equation}
314  \label{eq:scale_factors}
315  \begin{aligned}
316    e_1 &= (a + z) \lt[ \lt( \pd[\lambda]{i} \cos \varphi \rt)^2 + \lt( \pd[\varphi]{i} \rt)^2 \rt]^{1/2} \\
317    e_2 &= (a + z) \lt[ \lt( \pd[\lambda]{j} \cos \varphi \rt)^2 + \lt( \pd[\varphi]{j} \rt)^2 \rt]^{1/2} \\
318    e_3 &= \lt( \pd[z]{k} \rt)
319  \end{aligned}
320\end{equation}
321
322% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
323\begin{figure}[!tb]
324  \begin{center}
325    \includegraphics[]{Fig_I_earth_referential}
326    \caption{
327      \protect\label{fig:referential}
328      the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear
329      coordinate system $(i,j,k)$.
330    }
331  \end{center}
332\end{figure}
333%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
334
335Since the ocean depth is far smaller than the earth's radius, $a + z$, can be replaced by $a$ in
336(\autoref{eq:scale_factors}) (thin-shell approximation).
337The resulting horizontal scale factors $e_1$, $e_2$  are independent of $k$ while
338the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$.
339The scalar and vector operators that appear in the primitive equations
340(\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form,
341invariant in any orthogonal horizontal curvilinear coordinate system transformation:
342\begin{subequations}
343  % \label{eq:PE_discrete_operators}
344  \begin{gather}
345    \label{eq:PE_grad}
346    \nabla q =   \frac{1}{e_1} \pd[q]{i} \; \vect i
347               + \frac{1}{e_2} \pd[q]{j} \; \vect j
348               + \frac{1}{e_3} \pd[q]{k} \; \vect k \\
349    \label{eq:PE_div}
350    \nabla \cdot \vect A =   \frac{1}{e_1 \; e_2} \lt[ \pd[(e_2 \; a_1)]{\partial i} + \pd[(e_1 \; a_2)]{j} \rt]
351                           + \frac{1}{e_3} \lt[ \pd[a_3]{k} \rt]
352  \end{gather}
353  \begin{multline}
354    \label{eq:PE_curl}
355      \nabla \times \vect{A} =   \lt[ \frac{1}{e_2} \pd[a_3]{j} - \frac{1}{e_3} \pd[a_2]{k}   \rt] \vect i \\
356                               + \lt[ \frac{1}{e_3} \pd[a_1]{k} - \frac{1}{e_1} \pd[a_3]{i}   \rt] \vect j \\
357                               + \frac{1}{e_1 e_2} \lt[ \pd[(e_2 a_2)]{i} - \pd[(e_1 a_1)]{j} \rt] \vect k
358  \end{multline}
359  \begin{gather}
360    \label{eq:PE_lap}
361    \Delta q = \nabla \cdot (\nabla q) \\
362    \label{eq:PE_lap_vector}
363    \Delta \vect A = \nabla (\nabla \cdot \vect A) - \nabla \times (\nabla \times \vect A)
364  \end{gather}
365\end{subequations}
366where $q$ is a scalar quantity and $\vect A = (a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinates system.
367
368% -------------------------------------------------------------------------------------------------------------
369% Continuous Model Equations
370% -------------------------------------------------------------------------------------------------------------
371\subsection{Continuous model equations}
372\label{subsec:PE_zco_Eq}
373
374In order to express the Primitive Equations in tensorial formalism,
375it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using
376\autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}.
377Let us set $\vect U = (u,v,w) = \vect U_h + w \; \vect k $, the velocity in the $(i,j,k)$ coordinates system and
378define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
379\begin{gather}
380  \label{eq:PE_curl_Uh}
381  \zeta = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, v)]{i} - \pd[(e_1 \, u)]{j} \rt] \\
382  \label{eq:PE_div_Uh}
383  \chi  = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, u)]{i} + \pd[(e_1 \, v)]{j} \rt]
384\end{gather}
385
386Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that
387$e_3$  is a function of the single variable $k$,
388$NLT$ the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows:
389\begin{alignat*}{2}
390  &NLT &=   &\lt[ (\nabla \times {\vect U}) \times {\vect U} + \frac{1}{2} \nabla \lt( {\vect U}^2 \rt) \rt]_h \\
391  &    &=   &\lt(
392    \begin{array}{*{20}c}
393                \lt[ \frac{1}{e_3} \pd[u]{k} - \frac{1}{e_1} \pd[w]{i} \rt] w - \zeta \; v   \\
394                \zeta \; u - \lt[ \frac{1}{e_2} \pd[w]{j} - \frac{1}{e_3} \pd[v]{k} \rt] \ w
395    \end{array}
396                                                                                             \rt)
397          + \frac{1}{2} \lt(
398    \begin{array}{*{20}c}
399                             \frac{1}{e_1} \pd[(u^2 + v^2 + w^2)]{i} \\
400                             \frac{1}{e_2} \pd[(u^2 + v^2 + w^2)]{j}
401    \end{array}
402                                                                     \rt) \\
403  &    &=   &\lt(
404    \begin{array}{*{20}c}
405                  -\zeta \; v \\
406                   \zeta \; u
407    \end{array}
408                              \rt)
409          + \frac{1}{2} \lt(
410    \begin{array}{*{20}c}
411                             \frac{1}{e_1} \pd[(u^2 + v^2)]{i} \\
412                             \frac{1}{e_2} \pd[(u^2 + v^2)]{j}
413    \end{array}
414                                                               \rt) \\
415  &    &  &+ \frac{1}{e_3} \lt(
416    \begin{array}{*{20}c}
417                                w \; \pd[u]{k} \\
418                                w \; \pd[v]{k}
419    \end{array}
420                                               \rt)
421           - \lt(
422    \begin{array}{*{20}c}
423                  \frac{w}{e_1} \pd[w]{i} - \frac{1}{2 e_1} \pd[w^2]{i} \\
424                  \frac{w}{e_2} \pd[w]{j} - \frac{1}{2 e_2} \pd[w^2]{j}
425    \end{array}
426                                                                        \rt)
427\end{alignat*}
428The last term of the right hand side is obviously zero, and thus the nonlinear term of
429\autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system:
430\begin{equation}
431  \label{eq:PE_vector_form}
432  NLT =   \zeta \; \vect k \times \vect U_h + \frac{1}{2} \nabla_h \lt( \vect U_h^2 \rt)
433        + \frac{1}{e_3} w \pd[\vect U_h]{k}
434\end{equation}
435
436This is the so-called \textit{vector invariant form} of the momentum advection term.
437For some purposes, it can be advantageous to write this term in the so-called flux form,
438\ie to write it as the divergence of fluxes.
439For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows:
440\begin{alignat*}{2}
441  &NLT_i &= &- \zeta \; v + \frac{1}{2 \; e_1} \pd[ (u^2 + v^2) ]{i} + \frac{1}{e_3} w \ \pd[u]{k} \\
442  &      &&\frac{1}{e_1 \; e_2} \lt( -v \pd[(e_2 \, v)]{i} + v \pd[(e_1 \, u)]{j} \rt)
443            + \frac{1}{e_1 e_2} \lt( e_2 \; u \pd[u]{i} + e_2 \; v \pd[v]{i} \rt) \\
444  &      & &+ \frac{1}{e_3} \lt( w \; \pd[u]{k} \rt) \\
445  &      &&\frac{1}{e_1 \; e_2} \lt[ - \lt( v^2 \pd[e_2]{i} + e_2 \, v \pd[v]{i} \rt)
446                                     + \lt( \pd[ \lt( e_1 \, u \, v \rt)]{j} -         e_1 \, u \pd[v]{j} \rt) \rt. \\
447  &      &                       &\lt. + \lt( \pd[ \lt( e_2 \, u \, u \rt)]{i} - u \pd[ \lt( e_2 u \rt)]{i} \rt)
448                                     + e_2 v \pd[v]{i}                                                         \rt] \\
449  &      & &+ \frac{1}{e_3} \lt( \pd[(w \, u)]{k} - u \pd[w]{k} \rt) \\
450  &      &&\frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, u \, v)]{j} \rt)
451            + \frac{1}{e_3} \pd[(w \, u)]{k} \\
452  &      & &+ \frac{1}{e_1 e_2} \lt[ - u \lt( \pd[(e_1 v)]{j} - v \, \pd[e_1]{j} \rt)
453                                  - u \pd[(e_2 u)]{i}                              \rt]
454            - \frac{1}{e_3} \pd[w]{k} u \\
455  &      & &+ \frac{1}{e_1 e_2} \lt( - v^2 \pd[e_2]{i} \rt) \\
456  &      &= &\nabla \cdot (\vect U \, u) - (\nabla \cdot \vect U) \ u
457            + \frac{1}{e_1 e_2} \lt( -v^2 \pd[e_2]{i} + u v \, \pd[e_1]{j} \rt) \\
458  \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it comes:}
459  &      &= &\, \nabla \cdot (\vect U \, u) + \frac{1}{e_1 e_2} \lt( v \; \pd[e_2]{i} - u \; \pd[e_1]{j} \rt) (-v)
460\end{alignat*}
461
462The flux form of the momentum advection term is therefore given by:
463\begin{equation}
464  \label{eq:PE_flux_form}
465  NLT =   \nabla \cdot \lt(
466    \begin{array}{*{20}c}
467                            \vect U \, u \\
468                            \vect U \, v
469    \end{array}
470                                         \rt)
471        + \frac{1}{e_1 e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \vect k \times \vect U_h
472\end{equation}
473
474The flux form has two terms,
475the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation)
476and the second one is due to the curvilinear nature of the coordinate system used.
477The latter is called the \textit{metric} term and can be viewed as a modification of the Coriolis parameter:
478\[
479  % \label{eq:PE_cor+metric}
480  f \to f + \frac{1}{e_1 e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt)
481\]
482
483Note that in the case of geographical coordinate,
484\ie when $(i,j) \to (\lambda,\varphi)$ and $(e_1,e_2) \to (a \, \cos \varphi,a)$,
485we recover the commonly used modification of the Coriolis parameter $f \to f + (u / a) \tan \varphi$.
486
487To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be written in
488the following tensorial formalism:
489
490\begin{itemize}
491\item
492  \textbf{Vector invariant form of the momentum equations}:
493  \begin{equation}
494    \label{eq:PE_dyn_vect}
495    \begin{split}
496    % \label{eq:PE_dyn_vect_u}
497      \pd[u]{t} = &+ (\zeta + f) \, v - \frac{1}{2 e_1} \pd[]{i} (u^2 + v^2)
498                   - \frac{1}{e_3} w \pd[u]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\
499                  &+ D_u^{\vect U} + F_u^{\vect U} \\
500      \pd[v]{t} = &- (\zeta + f) \, u - \frac{1}{2 e_2} \pd[]{j} (u^2 + v^2)
501                   - \frac{1}{e_3} w \pd[v]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ 
502                  &+ D_v^{\vect U} + F_v^{\vect U}
503    \end{split}
504  \end{equation}
505\item
506  \textbf{flux form of the momentum equations}:
507  % \label{eq:PE_dyn_flux}
508  \begin{multline*}
509    % \label{eq:PE_dyn_flux_u}
510    \pd[u]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v \\
511                - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) \\
512                - \frac{1}{e_3} \pd[(w \, u)]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt)
513                + D_u^{\vect U} + F_u^{\vect U}
514  \end{multline*}
515  \begin{multline*}
516    % \label{eq:PE_dyn_flux_v}
517    \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\
518                + \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\
519                - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt)
520                + D_v^{\vect U} + F_v^{\vect U}
521  \end{multline*}
522  where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and $p_s$, the surface pressure,
523  is given by:
524  \[
525  % \label{eq:PE_spg}
526    p_s = \rho \,g \, \eta
527  \]
528  with $\eta$ is solution of \autoref{eq:PE_ssh}.
529
530  The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
531  \[
532  % \label{eq:w_diag}
533    \pd[w]{k} = - \chi \; e_3 \qquad 
534  % \label{eq:hp_diag}
535    \pd[p_h]{k} = - \rho \; g \; e_3
536  \]
537  where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}.
538\item \textit{tracer equations}:
539  \[
540    %\label{eq:S}
541    \pd[T]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt]
542                - \frac{1}{e_3} \pd[(T \, w)]{k} + D^T + F^T \\
543    %\label{eq:T}
544    \pd[S]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt]
545                - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S
546    %\label{eq:rho}
547    \rho = \rho \big( T,S,z(k) \big)
548  \]
549\end{itemize}
550
551The expression of $\vect D^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used.
552It will be defined in \autoref{eq:PE_zdf}.
553The nature and formulation of $\vect F^{\vect U}$, $F^T$ and $F^S$, the surface forcing terms,
554are discussed in \autoref{chap:SBC}.
555
556\newpage
557
558% ================================================================
559% Curvilinear generalised vertical coordinate System
560% ================================================================
561\section{Curvilinear generalised vertical coordinate system}
562\label{sec:PE_gco}
563
564The ocean domain presents a huge diversity of situation in the vertical.
565First the ocean surface is a time dependent surface (moving surface).
566Second the ocean floor depends on the geographical position,
567varying from more than 6,000 meters in abyssal trenches to zero at the coast.
568Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing.
569Therefore, in order to represent the ocean with respect to
570the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height
571\eg an \zstar-coordinate;
572for the second point, a space variation to fit the change of bottom topography
573\eg a terrain-following or $\sigma$-coordinate;
574and for the third point, one will be tempted to use a space and time dependent coordinate that
575follows the isopycnal surfaces, \eg an isopycnic coordinate.
576
577In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in
578HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at
579the ocean bottom) \citep{chassignet.smith.ea_JPO03} or
580OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere)
581\citep{madec.delecluse.ea_JPO96} among others.
582
583In fact one is totally free to choose any space and time vertical coordinate by
584introducing an arbitrary vertical coordinate :
585\begin{equation}
586  \label{eq:PE_s}
587  s = s(i,j,k,t)
588\end{equation}
589with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$,
590when $i$, $j$ and $t$ are held fixed.
591\autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into
592the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through
593\autoref{eq:PE_s}.
594This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact
595an Arbitrary Lagrangian--Eulerian (ALE) coordinate.
596Indeed, choosing an expression for $s$ is an arbitrary choice that determines
597which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and
598which part will be used to move them (Lagrangian part).
599The coordinate is also sometime referenced as an adaptive coordinate \citep{hofmeister.burchard.ea_OM10},
600since the coordinate system is adapted in the course of the simulation.
601Its most often used implementation is via an ALE algorithm,
602in which a pure lagrangian step is followed by regridding and remapping steps,
603the later step implicitly embedding the vertical advection
604\citep{hirt.amsden.ea_JCP74, chassignet.smith.ea_JPO03, white.adcroft.ea_JCP09}.
605Here we follow the \citep{kasahara_MWR74} strategy:
606a regridding step (an update of the vertical coordinate) followed by an eulerian step with
607an explicit computation of vertical advection relative to the moving s-surfaces.
608
609%\gmcomment{
610%A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient...
611the generalized vertical coordinates used in ocean modelling are not orthogonal,
612which contrasts with many other applications in mathematical physics.
613Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter.
614
615The horizontal velocity in ocean models measures motions in the horizontal plane,
616perpendicular to the local gravitational field.
617That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential,
618isopycnal, pressure, or terrain following.
619The key motivation for maintaining the same horizontal velocity component is that
620the hydrostatic and geostrophic balances are dominant in the large-scale ocean.
621Use of an alternative quasi -horizontal velocity, for example one oriented parallel to the generalized surface,
622would lead to unacceptable numerical errors.
623Correspondingly, the vertical direction is anti -parallel to the gravitational force in
624all of the coordinate systems.
625We do not choose the alternative of a quasi -vertical direction oriented normal to
626the surface of a constant generalized vertical coordinate.
627
628It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between
629the vertical coordinate choices.
630That is, computation of the dia-surface velocity component represents the fundamental distinction between
631the various coordinates.
632In some models, such as geopotential, pressure, and terrain following, this transport is typically diagnosed from
633volume or mass conservation.
634In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about
635the physical processes producing a flux across the layer interfaces.
636
637In this section we first establish the PE in the generalised vertical $s$-coordinate,
638then we discuss the particular cases available in \NEMO, namely $z$, \zstar, $s$, and \ztilde.
639%}
640
641% -------------------------------------------------------------------------------------------------------------
642% The s-coordinate Formulation
643% -------------------------------------------------------------------------------------------------------------
644\subsection{\textit{S}-coordinate formulation}
645
646Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k = z$ and
647thus $e_3 = 1$, we introduce an arbitrary vertical coordinate $s = s(i,j,k,t)$,
648which includes $z$-, \zstar- and $\sigma$-coordinates as special cases
649($s = z$, $s = \zstar$, and $s = \sigma = z / H$ or $ = z / \lt( H + \eta \rt)$, resp.).
650A formal derivation of the transformed equations is given in \autoref{apdx:A}.
651Let us define the vertical scale factor by $e_3 = \partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ),
652and the slopes in the $(i,j)$ directions between $s$- and $z$-surfaces by:
653\begin{equation}
654  \label{eq:PE_sco_slope}
655  \sigma_1 = \frac{1}{e_1} \; \lt. \pd[z]{i} \rt|_s \quad \text{and} \quad
656  \sigma_2 = \frac{1}{e_2} \; \lt. \pd[z]{j} \rt|_s
657\end{equation}
658We also introduce $\omega$, a dia-surface velocity component, defined as the velocity
659relative to the moving $s$-surfaces and normal to them:
660\[
661  % \label{eq:PE_sco_w}
662  \omega = w - e_3 \, \pd[s]{t} - \sigma_1 \, u - \sigma_2 \, v
663\]
664
665The equations solved by the ocean model \autoref{eq:PE} in $s$-coordinate can be written as follows
666(see \autoref{sec:A_momentum}):
667
668\begin{itemize}
669\item \textbf{Vector invariant form of the momentum equation}:
670  \begin{multline*}
671  % \label{eq:PE_sco_u_vector}
672    \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\
673                - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + g \frac{\rho}{\rho_o} \sigma_1
674                + D_u^{\vect U} + F_u^{\vect U}
675  \end{multline*}
676  \begin{multline*}
677  % \label{eq:PE_sco_v_vector}
678    \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\
679                - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + g \frac{\rho}{\rho_o} \sigma_2
680                + D_v^{\vect U} + F_v^{\vect U}
681  \end{multline*}
682\item \textbf{Flux form of the momentum equation}:
683  \begin{multline*}
684  % \label{eq:PE_sco_u_flux}
685    \frac{1}{e_3} \pd[(e_3 \, u)]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v \\
686                                       - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) \\
687                                       - \frac{1}{e_3} \pd[(\omega \, u)]{k}
688                                       - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt)
689                                       + g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U}
690  \end{multline*}
691  \begin{multline*}
692  % \label{eq:PE_sco_v_flux}
693    \frac{1}{e_3} \pd[(e_3 \, v)]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\
694                                       - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) \\
695                                       - \frac{1}{e_3} \pd[(\omega \, v)]{k}
696                                       - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt)
697                                       + g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U}
698  \end{multline*}
699  where the relative vorticity, $\zeta$, the surface pressure gradient,
700  and the hydrostatic pressure have the same expressions as in $z$-coordinates although
701  they do not represent exactly the same quantities.
702  $\omega$ is provided by the continuity equation (see \autoref{apdx:A}):
703  \[
704  % \label{eq:PE_sco_continuity}
705    \pd[e_3]{t} + e_3 \; \chi + \pd[\omega]{s} = 0 \quad \text{with} \quad
706    \chi = \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u)]{i} + \pd[(e_1 e_3 \, v)]{j} \rt)
707  \]
708\item \textit{tracer equations}:
709  \begin{multline*}
710  % \label{eq:PE_sco_t}
711    \frac{1}{e_3} \pd[(e_3 \, T)]{t} = - \frac{1}{e_1 e_2 e_3} \lt(   \pd[(e_2 e_3 \, u \, T)]{i}
712                                                                    + \pd[(e_1 e_3 \, v \, T)]{j} \rt) \\
713                                       - \frac{1}{e_3} \pd[(T \, \omega)]{k} + D^T + F^S
714  \end{multline*}
715  \begin{multline}
716  % \label{eq:PE_sco_s}
717    \frac{1}{e_3} \pd[(e_3 \, S)]{t} = - \frac{1}{e_1 e_2 e_3} \lt(   \pd[(e_2 e_3 \, u \, S)]{i}
718                                                                    + \pd[(e_1 e_3 \, v \, S)]{j} \rt) \\
719                                       - \frac{1}{e_3} \pd[(S \, \omega)]{k} + D^S + F^S
720  \end{multline}
721\end{itemize}
722The equation of state has the same expression as in $z$-coordinate,
723and similar expressions are used for mixing and forcing terms.
724
725\gmcomment{
726  \colorbox{yellow}{ to be updated $= = >$}
727  Add a few works on z and zps and s and underlies the differences between all of them
728  \colorbox{yellow}{$< = =$ end update}
729}
730
731% -------------------------------------------------------------------------------------------------------------
732% Curvilinear \zstar-coordinate System
733% -------------------------------------------------------------------------------------------------------------
734\subsection{Curvilinear \zstar-coordinate system}
735\label{subsec:PE_zco_star}
736
737%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
738\begin{figure}[!b]
739  \begin{center}
740    \includegraphics[]{Fig_z_zstar}
741    \caption{
742      \protect\label{fig:z_zstar}
743      (a) $z$-coordinate in linear free-surface case ;
744      (b) $z$-coordinate in non-linear free surface case ;
745      (c) re-scaled height coordinate
746      (become popular as the \zstar-coordinate \citep{adcroft.campin_OM04}).
747    }
748  \end{center}
749\end{figure}
750%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
751
752In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
753These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site.
754
755The \zstar coordinate approach is an unapproximated, non-linear free surface implementation which allows one to
756deal with large amplitude free-surface variations relative to the vertical resolution \citep{adcroft.campin_OM04}.
757In the \zstar formulation,
758the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level,
759as in the $z$-coordinate formulation, but is equally distributed over the full water column.
760Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth,
761as illustrated by \autoref{fig:z_zstar}.
762Note that with a flat bottom, such as in \autoref{fig:z_zstar}, the bottom-following $z$ coordinate and \zstar are equivalent.
763The definition and modified oceanic equations for the rescaled vertical coordinate \zstar,
764including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004).
765The major points are summarized here.
766The position (\zstar) and vertical discretization (\zstar) are expressed as:
767\[
768  % \label{eq:z-star}
769  H + \zstar = (H + z)  / r \quad \text{and}  \quad \delta \zstar
770              = \delta z / r \quad \text{with} \quad r
771              = \frac{H + \eta}{H}
772\]
773Since the vertical displacement of the free surface is incorporated in the vertical coordinate \zstar,
774the upper and lower boundaries are at fixed  \zstar position,
775$\zstar = 0$ and $\zstar = -H$ respectively.
776Also the divergence of the flow field is no longer zero as shown by the continuity equation:
777\[
778  \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) (r \; w *) = 0
779\]
780
781% from MOM4p1 documentation
782To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
783\[
784  % \label{eq:PE_}
785  \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt)
786\]
787
788This coordinate is closely related to the "eta" coordinate used in many atmospheric models
789(see Black (1994) for a review of eta coordinate atmospheric models).
790It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
791and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
792
793The surfaces of constant \zstar are quasi -horizontal.
794Indeed, the \zstar coordinate reduces to $z$ when $\eta$ is zero.
795In general, when noting the large differences between
796undulations of the bottom topography versus undulations in the surface height,
797it is clear that surfaces constant \zstar are very similar to the depth surfaces.
798These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
799terrain following sigma models discussed in \autoref{subsec:PE_sco}.
800Additionally, since \zstar when $\eta = 0$,
801no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography.
802This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
803nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
804depending on the sophistication of the pressure gradient solver.
805The quasi -horizontal nature of the coordinate surfaces also facilitates the implementation of
806neutral physics parameterizations in \zstar models using the same techniques as in $z$-models
807(see Chapters 13-16 of \cite{griffies_bk04}) for a discussion of neutral physics in $z$-models,
808as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
809
810The range over which \zstar varies is time independent $-H \leq \zstar \leq 0$.
811Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
812This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
813
814Because \zstar has a time independent range, all grid cells have static increments ds,
815and the sum of the ver tical increments yields the time independent ocean depth. %k ds = H.
816The \zstar coordinate is therefore invisible to undulations of the free surface,
817since it moves along with the free surface.
818This proper ty means that no spurious vertical transport is induced across surfaces of constant \zstar by
819the motion of external gravity waves.
820Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
821Quite generally, the time independent range for the \zstar coordinate is a very convenient property that
822allows for a nearly arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
823the surface height, again so long as $\eta > -H$.
824%end MOM doc %%%
825
826\newpage 
827
828% -------------------------------------------------------------------------------------------------------------
829% Terrain following  coordinate System
830% -------------------------------------------------------------------------------------------------------------
831\subsection{Curvilinear terrain-following \textit{s}--coordinate}
832\label{subsec:PE_sco}
833
834% -------------------------------------------------------------------------------------------------------------
835% Introduction
836% -------------------------------------------------------------------------------------------------------------
837\subsubsection{Introduction}
838
839Several important aspects of the ocean circulation are influenced by bottom topography.
840Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and
841channels that strongly constrain the path of water masses, but more subtle effects exist.
842For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes.
843Topographic Rossby waves can be excited and can interact with the mean current.
844In the $z$-coordinate system presented in the previous section (\autoref{sec:PE_zco}),
845$z$-surfaces are geopotential surfaces.
846The bottom topography is discretised by steps.
847This often leads to a misrepresentation of a gradually sloping bottom and to
848large localized depth gradients associated with large localized vertical velocities.
849The response to such a velocity field often leads to numerical dispersion effects.
850One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of
851a full step one \cite{pacanowski.gnanadesikan_MWR98}.
852Another solution is to introduce a terrain-following coordinate system (hereafter $s$-coordinate).
853
854The $s$-coordinate avoids the discretisation error in the depth field since the layers of
855computation are gradually adjusted with depth to the ocean bottom.
856Relatively small topographic features as well as  gentle, large-scale slopes of the sea floor in the deep ocean,
857which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths,
858can easily be represented (with relatively low vertical resolution).
859A terrain-following model (hereafter $s$-model) also facilitates the modelling of the boundary layer flows over
860a large depth range, which in the framework of the $z$-model would require high vertical resolution over
861the whole depth range.
862Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as
863the only boundaries of the domain (no more lateral boundary condition to specify).
864Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean,
865it has strong limitations as soon as stratification is introduced.
866The main two problems come from the truncation error in the horizontal pressure gradient and
867a possibly increased diapycnal diffusion.
868The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:A}),
869
870\begin{equation}
871  \label{eq:PE_p_sco}
872  \nabla p |_z = \nabla p |_s - \pd[p]{s} \nabla z |_s
873\end{equation}
874
875The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and
876introduces a truncation error that is not present in a $z$-model.
877In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$),
878\citet{haney_JPO91} and \citet{beckmann.haidvogel_JPO93} have given estimates of the magnitude of this truncation error.
879It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state,
880and the finite difference scheme.
881This error limits the possible topographic slopes that a model can handle at
882a given horizontal and vertical resolution.
883This is a severe restriction for large-scale applications using realistic bottom topography.
884The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive.
885This problem can be at least partially overcome by mixing $s$-coordinate and
886step-like representation of bottom topography \citep{gerdes_JGR93*a,gerdes_JGR93*b,madec.delecluse.ea_JPO96}.
887However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for
888a realistic bottom topography:
889a envelope topography is defined in $s$-coordinate on which a full or
890partial step bottom topography is then applied in order to adjust the model depth to the observed one
891(see \autoref{sec:DOM_zgr}.
892
893For numerical reasons a minimum of diffusion is required along the coordinate surfaces of
894any finite difference model.
895It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces.
896This is the case for a $z$-model as well as for a $s$-model.
897However, density varies more strongly on $s$-surfaces than on horizontal surfaces in regions of
898large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model.
899Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus
900the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation.
901For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification.
902Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography,
903and thus will generate there a baroclinic eddy.
904In contrast, the ocean will stay at rest in a $z$-model.
905As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below
906the strongly stratified portion of the water column (\ie the main thermocline) \citep{madec.delecluse.ea_JPO96}.
907An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces
908(see \autoref{subsec:PE_ldf}).
909Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large,
910strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}).
911
912The $s$-coordinates introduced here \citep{lott.madec.ea_OM90,madec.delecluse.ea_JPO96} differ mainly in two aspects from
913similar models:
914it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography;
915It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate.
916
917% -------------------------------------------------------------------------------------------------------------
918% Curvilinear z-tilde coordinate System
919% -------------------------------------------------------------------------------------------------------------
920\subsection{\texorpdfstring{Curvilinear \ztilde-coordinate}{}}
921\label{subsec:PE_zco_tilde}
922
923The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}.
924It is available in \NEMO since the version 3.4.
925Nevertheless, it is currently not robust enough to be used in all possible configurations.
926Its use is therefore not recommended.
927
928\newpage 
929
930% ================================================================
931% Subgrid Scale Physics
932% ================================================================
933\section{Subgrid scale physics}
934\label{sec:PE_zdf_ldf}
935
936The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than
937a few kilometres in the horizontal, a few meters in the vertical and a few minutes.
938They are usually solved at larger scales: the specified grid spacing and time step of the numerical model.
939The effects of smaller scale motions (coming from the advective terms in the Navier-Stokes equations)
940must be represented entirely in terms of large-scale patterns to close the equations.
941These effects appear in the equations as the divergence of turbulent fluxes
942(\ie fluxes associated with the mean correlation of small scale perturbations).
943Assuming a turbulent closure hypothesis is equivalent to choose a formulation for these fluxes.
944It is usually called the subgrid scale physics.
945It must be emphasized that this is the weakest part of the primitive equations,
946but also one of the most important for long-term simulations as
947small scale processes \textit{in fine} balance the surface input of kinetic energy and heat.
948
949The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions.
950Therefore subgrid-scale physics \textbf{D}$^{\vect U}$, $D^{S}$ and $D^{T}$  in
951\autoref{eq:PE_dyn}, \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into
952a lateral part \textbf{D}$^{l \vect U}$, $D^{l S}$ and $D^{l T}$ and
953a vertical part \textbf{D}$^{v \vect U}$, $D^{v S}$ and $D^{v T}$.
954The formulation of these terms and their underlying physics are briefly discussed in the next two subsections.
955
956% -------------------------------------------------------------------------------------------------------------
957% Vertical Subgrid Scale Physics
958% -------------------------------------------------------------------------------------------------------------
959\subsection{Vertical subgrid scale physics}
960\label{subsec:PE_zdf}
961
962The model resolution is always larger than the scale at which the major sources of vertical turbulence occur
963(shear instability, internal wave breaking...).
964Turbulent motions are thus never explicitly solved, even partially, but always parameterized.
965The vertical turbulent fluxes are assumed to depend linearly on the gradients of large-scale quantities
966(for example, the turbulent heat flux is given by $\overline{T' w'} = -A^{v T} \partial_z \overline T$,
967where $A^{v T}$ is an eddy coefficient).
968This formulation is analogous to that of molecular diffusion and dissipation.
969This is quite clearly a necessary compromise: considering only the molecular viscosity acting on
970large scale severely underestimates the role of turbulent diffusion and dissipation,
971while an accurate consideration of the details of turbulent motions is simply impractical.
972The resulting vertical momentum and tracer diffusive operators are of second order:
973\begin{equation}
974  \label{eq:PE_zdf}
975  \begin{gathered}
976    \vect D^{v \vect U} = \pd[]{z} \lt( A^{vm} \pd[\vect U_h]{z} \rt) \ , \\
977          D^{vT}       = \pd[]{z} \lt( A^{vT} \pd[T]{z}         \rt) \quad \text{and} \quad
978          D^{vS}       = \pd[]{z} \lt( A^{vT} \pd[S]{z}         \rt)
979  \end{gathered}
980\end{equation}
981where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively.
982At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified
983(see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}).
984All the vertical physics is embedded in the specification of the eddy coefficients.
985They can be assumed to be either constant, or function of the local fluid properties
986(\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency...),
987or computed from a turbulent closure model.
988The choices available in \NEMO are discussed in \autoref{chap:ZDF}).
989
990% -------------------------------------------------------------------------------------------------------------
991% Lateral Diffusive and Viscous Operators Formulation
992% -------------------------------------------------------------------------------------------------------------
993\subsection{Formulation of the lateral diffusive and viscous operators}
994\label{subsec:PE_ldf}
995
996Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies
997(which can be solved explicitly if the resolution is sufficient since
998their underlying physics are included in the primitive equations),
999and a sub mesoscale turbulence which is never explicitly solved even partially, but always parameterized.
1000The formulation of lateral eddy fluxes depends on whether the mesoscale is below or above the grid-spacing
1001(\ie the model is eddy-resolving or not).
1002
1003In non-eddy-resolving configurations, the closure is similar to that used for the vertical physics.
1004The lateral turbulent fluxes are assumed to depend linearly on the lateral gradients of large-scale quantities.
1005The resulting lateral diffusive and dissipative operators are of second order.
1006Observations show that lateral mixing induced by mesoscale turbulence tends to be along isopycnal surfaces
1007(or more precisely neutral surfaces \cite{mcdougall_JPO87}) rather than across them.
1008As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that
1009the `lateral' direction is the horizontal, \ie the lateral mixing is performed along geopotential surfaces.
1010This leads to a geopotential second order operator for lateral subgrid scale physics.
1011This assumption can be relaxed: the eddy-induced turbulent fluxes can be better approached by assuming that
1012they depend linearly on the gradients of large-scale quantities computed along neutral surfaces.
1013In such a case, the diffusive operator is an isoneutral second order operator and
1014it has components in the three space directions.
1015However,
1016both horizontal and isoneutral operators have no effect on mean (\ie large scale) potential energy whereas
1017potential energy is a main source of turbulence (through baroclinic instabilities).
1018\citet{gent.mcwilliams_JPO90} have proposed a parameterisation of mesoscale eddy-induced turbulence which
1019associates an eddy-induced velocity to the isoneutral diffusion.
1020Its mean effect is to reduce the mean potential energy of the ocean.
1021This leads to a formulation of lateral subgrid-scale physics made up of an isoneutral second order operator and
1022an eddy induced advective part.
1023In all these lateral diffusive formulations,
1024the specification of the lateral eddy coefficients remains the problematic point as
1025there is no really satisfactory formulation of these coefficients as a function of large-scale features.
1026
1027In eddy-resolving configurations, a second order operator can be used,
1028but usually the more scale selective biharmonic operator is preferred as
1029the grid-spacing is usually not small enough compared to the scale of the eddies.
1030The role devoted to the subgrid-scale physics is to dissipate the energy that
1031cascades toward the grid scale and thus to ensure the stability of the model while
1032not interfering with the resolved mesoscale activity.
1033Another approach is becoming more and more popular:
1034instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations,
1035one uses a advective scheme which is diffusive enough to maintain the model stability.
1036It must be emphasised that then, all the sub-grid scale physics is included in the formulation of
1037the advection scheme.
1038
1039All these parameterisations of subgrid scale physics have advantages and drawbacks.
1040There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are:
1041Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces,
1042\citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes.
1043For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces,
1044and UBS advection schemes when flux form is chosen for the momentum advection.
1045
1046\subsubsection{Lateral laplacian tracer diffusive operator}
1047
1048The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}):
1049\begin{equation}
1050  \label{eq:PE_iso_tensor}
1051  D^{lT} = \nabla \vect . \lt( A^{lT} \; \Re \; \nabla T \rt) \quad \text{with} \quad
1052  \Re =
1053    \begin{pmatrix}
1054      1    & 0    & -r_1          \\
1055      0    & 1    & -r_2          \\
1056      -r_1 & -r_2 & r_1^2 + r_2^2 \\
1057    \end{pmatrix}
1058\end{equation}
1059where $r_1$ and $r_2$ are the slopes between the surface along which the diffusive operator acts and
1060the model level (\eg $z$- or $s$-surfaces).
1061Note that the formulation \autoref{eq:PE_iso_tensor} is exact for
1062the rotation between geopotential and $s$-surfaces,
1063while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces.
1064Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{cox_OM87}.
1065First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and
1066dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity.
1067Second, the two isoneutral directions of diffusion are assumed to be independent since
1068the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:B}).
1069
1070For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero.
1071$\Re$ reduces to the identity in the horizontal direction, no rotation is applied.
1072
1073For \textit{geopotential} diffusion,
1074$r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces:
1075they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:PE_sco_slope}).
1076
1077For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces.
1078Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates.
1079In $z$-coordinates:
1080\begin{equation}
1081  \label{eq:PE_iso_slopes}
1082  r_1 = \frac{e_3}{e_1} \lt( \pd[\rho]{i} \rt) \lt( \pd[\rho]{k} \rt)^{-1} \quad
1083  r_2 = \frac{e_3}{e_2} \lt( \pd[\rho]{j} \rt) \lt( \pd[\rho]{k} \rt)^{-1}
1084\end{equation}
1085while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$.
1086
1087\subsubsection{Eddy induced velocity}
1088
1089When the \textit{eddy induced velocity} parametrisation (eiv) \citep{gent.mcwilliams_JPO90} is used,
1090an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
1091\[
1092  % \label{eq:PE_iso+eiv}
1093  D^{lT} = \nabla \cdot \lt( A^{lT} \; \Re \; \nabla T \rt) + \nabla \cdot \lt( \vect U^\ast \, T \rt)
1094\]
1095where $ \vect U^\ast = \lt( u^\ast,v^\ast,w^\ast \rt)$ is a non-divergent,
1096eddy-induced transport velocity. This velocity field is defined by:
1097\begin{gather}
1098  % \label{eq:PE_eiv}
1099  u^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_1 \rt) \\
1100  v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \\
1101  w^\ast = - \frac{1}{e_1 e_2} \lt[   \pd[]{i} \lt( A^{eiv} \; e_2 \, \tilde{r}_1 \rt)
1102                                     + \pd[]{j} \lt( A^{eiv} \; e_1 \, \tilde{r}_2 \rt) \rt]
1103\end{gather}
1104where $A^{eiv}$ is the eddy induced velocity coefficient
1105(or equivalently the isoneutral thickness diffusivity coefficient),
1106and $\tilde r_1$ and $\tilde r_2$ are the slopes between isoneutral and \textit{geopotential} surfaces.
1107Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate:
1108\begin{align}
1109  \label{eq:PE_slopes_eiv}
1110  \tilde{r}_n =
1111    \begin{cases}
1112      r_n            & \text{in $z$-coordinate}                \\
1113      r_n + \sigma_n & \text{in \zstar- and $s$-coordinates}
1114    \end{cases}
1115  \quad \text{where~} n = 1, 2
1116\end{align}
1117
1118The normal component of the eddy induced velocity is zero at all the boundaries.
1119This can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in the vicinity of
1120the boundaries.
1121The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}).
1122
1123\subsubsection{Lateral bilaplacian tracer diffusive operator}
1124
1125The lateral bilaplacian tracer diffusive operator is defined by:
1126\[
1127  % \label{eq:PE_bilapT}
1128  D^{lT}= - \Delta \; (\Delta T) \quad \text{where} \quad
1129  \Delta \bullet = \nabla \lt( \sqrt{B^{lT}} \; \Re \; \nabla \bullet \rt)
1130\]
1131It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with
1132the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
1133
1134\subsubsection{Lateral Laplacian momentum diffusive operator}
1135
1136The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by
1137applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}):
1138\begin{align*}
1139  % \label{eq:PE_lapU}
1140  \vect D^{l \vect U} &=   \nabla_h        \big( A^{lm}    \chi             \big)
1141                         - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\
1142                      &= \lt(   \frac{1}{e_1}     \pd[ \lt( A^{lm}    \chi      \rt) ]{i} \rt.
1143                              - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j}
1144                                \frac{1}{e_2}     \pd[ \lt( A^{lm}    \chi      \rt) ]{j}
1145                         \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt)
1146\end{align*}
1147
1148Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields
1149(see \autoref{apdx:C}).
1150Unfortunately, it is only available in \textit{iso-level} direction.
1151When a rotation is required
1152(\ie geopotential diffusion in $s$-coordinates or isoneutral diffusion in both $z$- and $s$-coordinates),
1153the $u$ and $v$-fields are considered as independent scalar fields, so that the diffusive operator is given by:
1154\begin{gather*}
1155  % \label{eq:PE_lapU_iso}
1156    D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \\
1157    D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt)
1158\end{gather*}
1159where $\Re$ is given by \autoref{eq:PE_iso_tensor}.
1160It is the same expression as those used for diffusive operator on tracers.
1161It must be emphasised that such a formulation is only exact in a Cartesian coordinate system,
1162\ie on a $f$- or $\beta$-plane, not on the sphere.
1163It is also a very good approximation in vicinity of the Equator in
1164a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}.
1165
1166\subsubsection{lateral bilaplacian momentum diffusive operator}
1167
1168As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with
1169the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
1170Nevertheless it is currently not available in the iso-neutral case.
1171
1172\biblio
1173
1174\pindex
1175
1176\end{document}
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