chap_model_basics.tex

all the sub-grid scale physics is included in the formulation of the advection scheme.

All these parameterisations of subgrid scale physics have advantages and drawbacks.
They are not all available in \NEMO.
For active tracers (temperature and salinity) the main ones are:
Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces,
\citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes.
For momentum, the main ones are:
Laplacian and bilaplacian operators acting along geopotential surfaces,
and UBS advection schemes when flux form is chosen for the momentum advection.

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\subsubsection{Lateral laplacian tracer diffusive operator}

The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:DIFFOPERS}):
\begin{equation}
  \label{eq:MB_iso_tensor}
  D^{lT} = \nabla \vect . \lt( A^{lT} \; \Re \; \nabla T \rt) \quad \text{with} \quad \Re =
  \begin{pmatrix}
    1    & 0    & -r_1          \\
    0    & 1    & -r_2          \\
    -r_1 & -r_2 & r_1^2 + r_2^2 \\
  \end{pmatrix}
\end{equation}
where $r_1$ and $r_2$ are the slopes between the surface along which the diffusive operator acts and
the model level (\eg\ $z$- or $s$-surfaces).
Note that the formulation \autoref{eq:MB_iso_tensor} is exact for
the rotation between geopotential and $s$-surfaces,
while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces.
Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:MB_iso_tensor}
\citep{cox_OM87}.
First, the horizontal contribution of the dianeutral mixing is neglected since
the ratio between iso and dia-neutral diffusive coefficients is known to be
several orders of magnitude smaller than unity.
Second, the two isoneutral directions of diffusion are assumed to be independent since
the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:DIFFOPERS}).

For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero.
$\Re$ reduces to the identity in the horizontal direction, no rotation is applied.

For \textit{geopotential} diffusion,
$r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces:
they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:MB_sco_slope}).

For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between
the isoneutral and computational surfaces.
Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates.
In $z$-coordinates:
\begin{equation}
  \label{eq:MB_iso_slopes}
  r_1 = \frac{e_3}{e_1} \lt( \pd[\rho]{i} \rt) \lt( \pd[\rho]{k} \rt)^{-1} \quad
  r_2 = \frac{e_3}{e_2} \lt( \pd[\rho]{j} \rt) \lt( \pd[\rho]{k} \rt)^{-1}
\end{equation}
while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$.

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\subsubsection{Eddy induced velocity}

When the \textit{eddy induced velocity} parametrisation (eiv) \citep{gent.mcwilliams_JPO90} is used,
an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
\[
  % \label{eq:MB_iso+eiv}
  D^{lT} = \nabla \cdot \lt( A^{lT} \; \Re \; \nabla T \rt) + \nabla \cdot \lt( \vect U^\ast \, T \rt)
\]
where $ \vect U^\ast = \lt( u^\ast,v^\ast,w^\ast \rt)$ is a non-divergent,
eddy-induced transport velocity. This velocity field is defined by:
\[
  % \label{eq:MB_eiv}
  u^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_1 \rt) \quad
  v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \quad
  w^\ast = - \frac{1}{e_1 e_2} \lt[   \pd[]{i} \lt( A^{eiv} \; e_2 \, \tilde{r}_1 \rt)
                                     + \pd[]{j} \lt( A^{eiv} \; e_1 \, \tilde{r}_2 \rt) \rt]
\]
where $A^{eiv}$ is the eddy induced velocity coefficient
(or equivalently the isoneutral thickness diffusivity coefficient),
and $\tilde r_1$ and $\tilde r_2$ are the slopes between
isoneutral and \textit{geopotential} surfaces.
Their values are thus independent of the vertical coordinate,
but their expression depends on the coordinate:
\begin{equation}
  \label{eq:MB_slopes_eiv}
  \tilde{r}_n =
    \begin{cases}
      r_n            & \text{in $z$-coordinate}                \\
      r_n + \sigma_n & \text{in \zstar- and $s$-coordinates}
    \end{cases}
  \quad \text{where~} n = 1, 2
\end{equation}

The normal component of the eddy induced velocity is zero at all the boundaries.
This can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in
the vicinity of the boundaries.
The latter strategy is used in \NEMO\ (cf. \autoref{chap:LDF}).

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\subsubsection{Lateral bilaplacian tracer diffusive operator}

The lateral bilaplacian tracer diffusive operator is defined by:
\[
  % \label{eq:MB_bilapT}
  D^{lT}= - \Delta \; (\Delta T) \quad \text{where} \quad
  \Delta \bullet = \nabla \lt( \sqrt{B^{lT}} \; \Re \; \nabla \bullet \rt)
\]
It is the Laplacian operator given by \autoref{eq:MB_iso_tensor} applied twice with
the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.

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\subsubsection{Lateral Laplacian momentum diffusive operator}

The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by
applying \autoref{eq:MB_lap_vector} to the horizontal velocity vector (see \autoref{apdx:DIFFOPERS}):
\begin{align*}
  % \label{eq:MB_lapU}
  \vect D^{l \vect U} &=   \nabla_h        \big( A^{lm}    \chi             \big)
                         - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\
                      &= \lt(   \frac{1}{e_1}     \pd[ \lt( A^{lm}    \chi      \rt) ]{i} \rt.
                              - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} ,
                                \frac{1}{e_2}     \pd[ \lt( A^{lm}    \chi      \rt) ]{j}
                         \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt)
\end{align*}

Such a formulation ensures a complete separation between
the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}).
Unfortunately, it is only available in \textit{iso-level} direction.
When a rotation is required
(\ie\ geopotential diffusion in $s$-coordinates or
isoneutral diffusion in both $z$- and $s$-coordinates),
the $u$ and $v$-fields are considered as independent scalar fields,
so that the diffusive operator is given by:
\[
  % \label{eq:MB_lapU_iso}
    D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \quad
    D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt)
\]
where $\Re$ is given by \autoref{eq:MB_iso_tensor}.
It is the same expression as those used for diffusive operator on tracers.
It must be emphasised that such a formulation is only exact in a Cartesian coordinate system,
\ie\ on a $f$- or $\beta$-plane, not on the sphere.
It is also a very good approximation in vicinity of the Equator in
a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}.

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\subsubsection{Lateral bilaplacian momentum diffusive operator}

As for tracers,
the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with
the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
Nevertheless it is currently not available in the iso-neutral case.

\subinc{\input{../../global/epilogue}}

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