apdx_invariants.tex

\documentclass[../main/NEMO_manual]{subfiles}

\begin{document}

\chapter{Discrete Invariants of the Equations}
\label{apdx:INVARIANTS}

\chaptertoc

\paragraph{Changes record} ~\\

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    {\em   3.6} & {\em ...} & {\em ...} \\
    {\em   3.4} & {\em ...} & {\em ...} \\
    {\em <=3.4} & {\em ...} & {\em ...}
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\clearpage

%%%  Appendix put in cmtgm as it has not been updated for \zstar and s coordinate
%I'm writting this appendix. It will be available in a forthcoming release of the documentation

%\cmtgm{

%% =================================================================================================
\section{Introduction / Notations}
\label{sec:INVARIANTS_0}

Notation used in this appendix in the demonstations:

fluxes at the faces of a $T$-box:
\[
  U = e_{2u}\,e_{3u}\; u  \qquad  V = e_{1v}\,e_{3v}\; v  \qquad W = e_{1w}\,e_{2w}\; \omega
\]

volume of cells at $u$-, $v$-, and $T$-points:
\[
  b_u = e_{1u}\,e_{2u}\,e_{3u}  \qquad  b_v = e_{1v}\,e_{2v}\,e_{3v}  \qquad b_t = e_{1t}\,e_{2t}\,e_{3t}
\]

partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$

$dv=e_1\,e_2\,e_3 \,di\,dj\,dk$  is the volume element, with only $e_3$ that depends on time.
$D$ and $S$ are the ocean domain volume and surface, respectively.
No wetting/drying is allow (\ie\ $\frac{\partial S}{\partial t} = 0$).
Let $k_s$ and $k_b$ be the ocean surface and bottom, resp.
(\ie\ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).
\begin{flalign*}
  z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s}  e_3(\tilde{k}) \;d\tilde{k}
  = \eta - \int\limits_k^{k_s}  e_3 \;d\tilde{k}
\end{flalign*}

Continuity equation with the above notation:
\[
  \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t}  \biggl\{  \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0
\]

A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when:
\[
  \partial_t \left( \int_D{ Q\;dv } \right) =0
\]
Noting that the coordinate system used ....  blah blah
\[
  \partial_t \left( \int_D {Q\;dv} \right) =  \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }
  =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0
\]
equation of evolution of $Q$ written as
the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form,
the quadratic quantity $\frac{1}{2}Q^2$ is conserved when:
\begin{flalign*}
  \partial_t \left(   \int_D{ \frac{1}{2} \,Q^2\;dv }   \right)
  =&  \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\
  =&  \int_D {         Q   \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }
  -  \int_D { \frac{1}{2} Q^2 \,\partial_t  (e_3) \;e_1e_2\;di\,dj\,dk } \\
\end{flalign*}
that is in a more compact form :
\begin{flalign}
  \label{eq:INVARIANTS_Q2_flux}
  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
  =&                   \int_D { \frac{Q}{e_3}  \partial_t \left( e_3 \, Q \right) dv }
  -   \frac{1}{2} \int_D {  \frac{Q^2}{e_3} \partial_t (e_3) \;dv }
\end{flalign}
equation of evolution of $Q$ written as the time evolution of $Q$ like for momentum in vector invariant form,
the quadratic quantity $\frac{1}{2}Q^2$ is conserved when:
\begin{flalign*}
  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
  =&  \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\
  =& \int_D {         Q      \partial_t Q  \;e_1e_2e_3\;di\,dj\,dk }
  +  \int_D { \frac{1}{2} Q^2 \, \partial_t e_3  \;e_1e_2\;di\,dj\,dk } \\
\end{flalign*}
that is in a more compact form:
\begin{flalign}
  \label{eq:INVARIANTS_Q2_vect}
  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
  =& \int_D {         Q   \,\partial_t Q  \;dv }
  +   \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv }
\end{flalign}

%% =================================================================================================
\section{Continuous conservation}
\label{sec:INVARIANTS_1}

The discretization of pimitive equation in $s$-coordinate (\ie\ time and space varying vertical coordinate)
must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy.

Let us first establish those constraint in the continuous world.
The total energy (\ie\ kinetic plus potential energies) is conserved:
\begin{flalign}
  \label{eq:INVARIANTS_Tot_Energy}
  \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 +  \rho \, g \, z \right) \;dv \right)  = & 0
\end{flalign}
under the following assumptions: no dissipation, no forcing (wind, buoyancy flux, atmospheric pressure variations),
mass conservation, and closed domain.

This equation can be transformed to obtain several sub-equalities.
The transformation for the advection term depends on whether the vector invariant form or
the flux form is used for the momentum equation.
Using \autoref{eq:INVARIANTS_Q2_vect} and introducing \autoref{eq:SCOORD_dyn_vect} in
\autoref{eq:INVARIANTS_Tot_Energy} for the former form and
using \autoref{eq:INVARIANTS_Q2_flux} and introducing \autoref{eq:SCOORD_dyn_flux} in
\autoref{eq:INVARIANTS_Tot_Energy} for the latter form leads to:

% \label{eq:INVARIANTS_E_tot}
advection term (vector invariant form):
\[
  % \label{eq:INVARIANTS_E_tot_vect_vor_1}
  \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
\]
\[
  % \label{eq:INVARIANTS_E_tot_vect_adv_1}
  \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv
  + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv
  -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0
\]
advection term (flux form):
\[
  % \label{eq:INVARIANTS_E_tot_flux_metric}
  \int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\;
  \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0
\]
\[
  % \label{eq:INVARIANTS_E_tot_flux_adv}
  \int\limits_D \textbf{U}_h \cdot     \left(                 {{
        \begin{array} {*{20}c}
          \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
          \nabla \cdot \left( \textbf{U}\,v \right) \hfill
        \end{array}}
    }           \right)   \;dv
  +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0
\]
coriolis term
\[
  % \label{eq:INVARIANTS_E_tot_cor}
  \int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0
\]
pressure gradient:
\[
  % \label{eq:INVARIANTS_E_tot_pg_1}
  - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
  + \int\limits_D g\, \rho \; \partial_t z  \;dv
\]

where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces.

blah blah....

The prognostic ocean dynamics equation can be summarized as follows:
\[
  \text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
  {\text{COR} + \text{ADV}                       }
  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
\]

Vector invariant form:
% \label{eq:INVARIANTS_E_tot_vect}
\[
  % \label{eq:INVARIANTS_E_tot_vect_vor_2}
  \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0
\]
\[
  % \label{eq:INVARIANTS_E_tot_vect_adv_2}
  \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv
  + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv
  -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0
\]
\[
  % \label{eq:INVARIANTS_E_tot_pg_2}
  - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv
  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
  + \int\limits_D g\, \rho \; \partial_t z  \;dv
\]

Flux form:
\begin{subequations}