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\begin{document}

% ================================================================
% Invariant of the Equations
% ================================================================
\chapter{Invariants of the Primitive Equations}
\label{chap:CONS}

\chaptertoc

The continuous equations of motion have many analytic properties.
Many quantities (total mass, energy, enstrophy, etc.) are strictly conserved in the inviscid and unforced limit,
while ocean physics conserve the total quantities on which they act (momentum, temperature, salinity) but
dissipate their total variance (energy, enstrophy, etc.).
Unfortunately, the finite difference form of these equations is not guaranteed to
retain all these important properties.
In constructing the finite differencing schemes, we wish to ensure that
certain integral constraints will be maintained.
In particular, it is desirable to construct the finite difference equations so that
horizontal kinetic energy and/or potential enstrophy of horizontally non-divergent flow,
and variance of temperature and salinity will be conserved in the absence of dissipative effects and forcing.
\citet{arakawa_JCP66} has first pointed out the advantage of this approach.
He showed that if integral constraints on energy are maintained,
the computation will be free of the troublesome "non linear" instability originally pointed out by
\citet{phillips_TAMS59}.
A consistent formulation of the energetic properties is also extremely important in carrying out
long-term numerical simulations for an oceanographic model.
Such a formulation avoids systematic errors that accumulate with time \citep{bryan_JCP97}.

The general philosophy of OPA which has led to the discrete formulation presented in {\S}II.2 and II.3 is to
choose second order non-diffusive scheme for advective terms for both dynamical and tracer equations.
At this level of complexity, the resulting schemes are dispersive schemes.
Therefore, they require the addition of a diffusive operator to be stable.
The alternative is to use diffusive schemes such as upstream or flux corrected schemes.
This last option was rejected because we prefer a complete handling of the model diffusion,
\ie\ of the model physics rather than letting the advective scheme produces its own implicit diffusion without
controlling the space and time structure of this implicit diffusion.
Note that in some very specific cases as passive tracer studies, the positivity of the advective scheme is required.
In that case, and in that case only, the advective scheme used for passive tracer is a flux correction scheme
\citep{Marti1992?, Levy1996?, Levy1998?}.

% -------------------------------------------------------------------------------------------------------------
%       Conservation Properties on Ocean Dynamics
% -------------------------------------------------------------------------------------------------------------
\section{Conservation properties on ocean dynamics}
\label{sec:CONS_Invariant_dyn}

The non linear term of the momentum equations has been split into a vorticity term,
a gradient of horizontal kinetic energy and a vertical advection term.
Three schemes are available for the former (see {\S}~II.2) according to the CPP variable defined
(default option\textbf{?}or \textbf{key{\_}vorenergy} or \textbf{key{\_}vorcombined} defined).
They differ in their conservative properties (energy or enstrophy conserving scheme).
The two latter terms preserve the total kinetic energy:
the large scale kinetic energy is also preserved in practice.
The remaining non-diffusive terms of the momentum equation
(namely the hydrostatic and surface pressure gradient terms) also preserve the total kinetic energy and
have no effect on the vorticity of the flow.

\textbf{* relative, planetary and total vorticity term:}

Let us define as either the relative, planetary and total potential vorticity, \ie, ?, and ?, respectively.
The continuous formulation of the vorticity term satisfies following integral constraints:
\[
  % \label{eq:CONS_vor_vorticity}
  \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma
        \;{\mathrm {\mathbf k}}\times {\textbf {U}}_h } \right)\;dv} =0
\]

\[
  % \label{eq:CONS_vor_enstrophy}
  if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot
    \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv}
  =0
\]

\[
  % \label{eq:CONS_vor_energy}
  \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
\]
where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element.
(II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part.
It means that $\varsigma^2$ is conserved for a horizontally non-divergent flow.
(II.4.1c) is even satisfied locally since the vorticity term is orthogonal to the horizontal velocity.
It means that the vorticity term has no contribution to the evolution of the total kinetic energy.
(II.4.1a) is obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied simultaneously with
a second order scheme.
Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11),
it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c),
while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C).
Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of
the vorticity term.

\textbf{* Gradient of kinetic energy / vertical advection}

In continuous formulation, the gradient of horizontal kinetic energy has no contribution to the evolution of
the vorticity as the curl of a gradient is zero.
This property is satisfied locally with the discrete form of both the gradient and the curl operator we have made
(property (II.1.9)~).
Another continuous property is that the change of horizontal kinetic energy due to
vertical advection is exactly balanced by the change of horizontal kinetic energy due to
the horizontal gradient of horizontal kinetic energy:

\begin{equation} \label{eq:CONS_keg_zad}
\int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial
{\textbf{U}}_h }{\partial k}\;dv}
\end{equation}

Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of
the mean and difference operators, \autoref{eq:CONS_keg_zad} is demonstrated in the Appendix C.
The main point here is that satisfying \autoref{eq:CONS_keg_zad} links the choice of the discrete forms of
the vertical advection and of the horizontal gradient of horizontal kinetic energy.
Choosing one imposes the other.
The discrete form of the vertical advection given in {\S}II.2-a is a direct consequence of
formulating the horizontal kinetic energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in
the gradient term.

\textbf{* hydrostatic pressure gradient term}

In continuous formulation, a pressure gradient has no contribution to the evolution of the vorticity as
the curl of a gradient is zero.
This properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~).
In addition, when the equation of state is linear
(\ie\ when an advective-diffusive equation for density can be derived from those of temperature and salinity)
the change of horizontal kinetic energy due to the work of pressure forces is balanced by the change of
potential energy due to buoyancy forces:

\[
  % \label{eq:CONS_hpg_pe}
  \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
\]

Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of
the mean and difference operators, (II.4.3) is demonstrated in the Appendix C.
The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of
$T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$,
is defined as the sum the vertical scale factors at $w$-points starting from the surface.

\textbf{* surface pressure gradient term}

In continuous formulation, the surface pressure gradient has no contribution to the evolution of vorticity.
This properties is trivially satisfied locally as (II.2.3)
(the equation verified by $\psi$ has been derived from the discrete formulation of the momentum equations,
vertical sum and curl).
Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5),
thus the property is only satisfied with the accuracy required on the solver.
In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of
surface pressure forces is exactly zero:
\[
  % \label{eq:CONS_spg}
  \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
\]

(II.4.4) is satisfied in discrete form only if
the discrete barotropic streamfunction time evolution equation is given by (II.2.3) (see appendix C).
This shows that (II.2.3) is the only way to compute the streamfunction,
otherwise there is no guarantee that the surface pressure force work vanishes.

% -------------------------------------------------------------------------------------------------------------
%       Conservation Properties on Ocean Thermodynamics
% -------------------------------------------------------------------------------------------------------------
\section{Conservation properties on ocean thermodynamics}
\label{sec:CONS_Invariant_tra}

In continuous formulation, the advective terms of the tracer equations conserve the tracer content and
the quadratic form of the tracer, \ie
\[
  % \label{eq:CONS_tra_tra2}
  \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
  \;\text{and}
  \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
\]

The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies
(II.4.5) (see appendix C).
Note that in both continuous and discrete formulations, there is generally no strict conservation of mass,
since the equation of state is non linear with respect to $T$ and $S$.
In practice, the mass is conserved with a very good accuracy.

% -------------------------------------------------------------------------------------------------------------
%       Conservation Properties on Momentum Physics
% -------------------------------------------------------------------------------------------------------------
\subsection{Conservation properties on momentum physics}
\label{subsec:CONS_Invariant_dyn_physics}

\textbf{* lateral momentum diffusion term}

The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~:
\[
  % \label{eq:CONS_dynldf_dyn}
  \int\limits_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left[ {\nabla
        _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta
            \;{\mathrm {\mathbf k}}} \right)} \right]\;dv} =0
\]

\[
  % \label{eq:CONS_dynldf_div}
  \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)}
    \right]\;dv} =0
\]

\[
  % \label{eq:CONS_dynldf_curl}
  \int_D {{\mathrm {\mathbf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)}
    \right]\;dv} \leqslant 0
\]

\[
  % \label{eq:CONS_dynldf_curl2}
  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\mathrm {\mathbf k}}\cdot
    \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h
        \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} \right]\;dv}
  \leqslant 0
\]

\[
  % \label{eq:CONS_dynldf_div2}
  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[
      {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left(
          {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} \right]\;dv} \leqslant 0
\]


(II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and
the divergence of the flow, while Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy and
the square of the divergence.
The two latter properties are only satisfied when the eddy coefficients are horizontally uniform.

Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of the mean and difference operators,
it is shown that the discrete form of the lateral momentum diffusion given in
{\S}II.2-c satisfies all the integral constraints (II.4.6) (see appendix C).
In particular, when the eddy coefficients are horizontally uniform,
a complete separation of vorticity and horizontal divergence fields is ensured,
so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence
(variance of the horizontal divergence) and \textit{vice versa}.
When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken,
the term associated to the horizontal gradient of the divergence is zero locally.
When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken,
the term associated to the vertical curl of the vorticity is zero locally.
The resulting term conserves $\chi$ and dissipates $\chi^2$ when the eddy coefficient is horizontally uniform.

\textbf{* vertical momentum diffusion term}

As for the lateral momentum physics, the continuous form of the vertical diffusion of
momentum satisfies following integral constraints~:

conservation of momentum, dissipation of horizontal kinetic energy

\[
  % \label{eq:CONS_dynzdf_dyn}
  \begin{aligned}
    & \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\
    & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\
  \end{aligned}
\]
conservation of vorticity, dissipation of enstrophy
\[
  % \label{eq:CONS_dynzdf_vor}
  \begin{aligned}
    & \int_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left( {\frac{1}{e_3
          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\mathrm
                  {\mathbf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\
    & \int_D {\zeta \,{\mathrm {\mathbf k}}\cdot \nabla \times \left( {\frac{1}{e_3
          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\mathrm
                  {\mathbf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\
  \end{aligned}
\]
conservation of horizontal divergence, dissipation of square of the horizontal divergence
\[
  % \label{eq:CONS_dynzdf_div}
  \begin{aligned}
    &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}}
          \right)} \right)\;dv} =0 \\
    & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}}
          \right)} \right)\;dv} \leq 0 \\
  \end{aligned}
\]

In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C).
In $s$-coordinates, only first order properties can be demonstrated,
\ie\ the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence.

% -------------------------------------------------------------------------------------------------------------
%       Conservation Properties on Tracer Physics
% -------------------------------------------------------------------------------------------------------------
\subsection{Conservation properties on tracer physics}
\label{subsec:CONS_Invariant_tra_physics}

The numerical schemes used for tracer subgridscale physics are written in such a way that
the heat and salt contents are conserved (equations in flux form, second order centred finite differences).
As a form flux is used to compute the temperature and salinity,
the quadratic form of these quantities (\ie\ their variance) globally tends to diminish.
As for the advective term, there is generally no strict conservation of mass even if,
in practice, the mass is conserved with a very good accuracy.

\textbf{* lateral physics: }conservation of tracer, dissipation of tracer
variance, i.e.

\[
  % \label{eq:CONS_traldf_t_t2}
  \begin{aligned}
    &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\
    &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\
  \end{aligned}
\]

\textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, \ie

\[
  % \label{eq:CONS_trazdf_t_t2}
  \begin{aligned}
    & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\
    & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\
  \end{aligned}
\]

Using the symmetry or anti-symmetry properties of the mean and difference operators,
it is shown that the discrete form of tracer physics given in {\S}~II.2-c satisfies all the integral constraints
(II.4.8) and (II.4.9) except the dissipation of the square of the tracer when non-geopotential diffusion is used
(see appendix C).
A discrete form of the lateral tracer physics can be derived which satisfies these last properties.
Nevertheless, it requires a horizontal averaging of the vertical component of the lateral physics that
prevents the use of implicit resolution in the vertical.
It has not been implemented.

\biblio

\pindex

\end{document}
