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