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8679f9097b Jeff* .. _sub_phys_pkg_gmredi:
                 
7337b40cfd Navi* GMREDI: Gent-McWilliams/Redi Eddy Parameterization
                 **************************************************
8679f9097b Jeff* 
7337b40cfd Navi* Introduction
                 ============
8679f9097b Jeff* 
7337b40cfd Navi* Package :filelink:`gmredi <pkg/gmredi>` parameterizes
                 the effects of unresolved mesoscale eddies on tracer distribution,
                 i.e., temperature, salinity and other tracers.
                 
                 There are two parts to the Redi/GM subgrid-scale parameterization of geostrophic
                 eddies. The first, the Redi scheme (Redi 1982 :cite:`redi1982`), aims to mix tracer properties along
8679f9097b Jeff* isentropes (neutral surfaces) by means of a diffusion operator oriented
a4576c7cde Juli* along the local isentropic surface. The second part, GM
7337b40cfd Navi* (Gent and McWiliams 1990 :cite:`gen-mcw:90`, Gent et al. 1995 :cite:`gen-eta:95`), adiabatically
                 rearranges tracers through an advective flux where the advecting flow
8679f9097b Jeff* is a function of slope of the isentropic surfaces.
                 
7337b40cfd Navi* A description of key package equations used is
                 given in :numref:`ssub_phys_pkg_gmredi_descr`.
                 CPP options enable or disable different aspects of the package
                 (:numref:`ssub_phys_pkg_gmredi_config`). Run-time options, flags, and filenames
                 are set in ``data.gmredi``
                 (:numref:`ssub_phys_pkg_gmredi_runtime`). Available diagnostics
                 output is listed in :numref:`ssub_phys_pkg_gmredi_diagnostics`.
                 
                 .. _ssub_phys_pkg_gmredi_descr:
                 
                 Description
                 ===========
                 
                 The first GCM implementation of the Redi scheme was by Cox (1987) :cite:`cox87` in the GFDL ocean
8679f9097b Jeff* circulation model. The original approach failed to distinguish between
                 isopycnals and surfaces of locally referenced potential density (now
7337b40cfd Navi* called neutral surfaces), which are proper isentropes for the ocean. As
8679f9097b Jeff* will be discussed later, it also appears that the Cox implementation is
                 susceptible to a computational mode. Due to this mode, the Cox scheme
                 requires a background lateral diffusion to be present to conserve the
                 integrity of the model fields.
                 
                 The GM parameterization was then added to the GFDL code in the form of a
7337b40cfd Navi* non-divergent bolus velocity. The method defines two streamfunctions
8679f9097b Jeff* expressed in terms of the isoneutral slopes subject to the boundary
                 condition of zero value on upper and lower boundaries. The horizontal
                 bolus velocities are then the vertical derivative of these functions.
7337b40cfd Navi* Here in lies a problem highlighted by Griffies et al. (1998) :cite:`gretal:98`: the
                 bolus velocities involve multiple derivatives on the potential density field,
                 which can consequently give rise to noise. Griffies et al. point out that the GM
8679f9097b Jeff* bolus fluxes can be identically written as a skew flux which involves
                 fewer differential operators. Further, combining the skew flux
                 formulation and Redi scheme, substantial cancellations take place to the
                 point that the horizontal fluxes are unmodified from the lateral
                 diffusion parameterization.
                 
b938a3c63b antn* .. _GM_redi_desc:
                 
8679f9097b Jeff* Redi scheme: Isopycnal diffusion
7337b40cfd Navi* --------------------------------
8679f9097b Jeff* 
                 The Redi scheme diffuses tracers along isopycnals and introduces a term
                 in the tendency (rhs) of such a tracer (here :math:`\tau`) of the form:
                 
7337b40cfd Navi* .. math:: \nabla \cdot ( \kappa_\rho {\bf K}_{\rm Redi} \nabla \tau )
8679f9097b Jeff* 
                 where :math:`\kappa_\rho` is the along isopycnal diffusivity and
7337b40cfd Navi* :math:`{\bf K}_{\rm Redi}` is a rank 2 tensor that projects the gradient of
8679f9097b Jeff* :math:`\tau` onto the isopycnal surface. The unapproximated projection
                 tensor is:
                 
                 .. math::
                 
a4576c7cde Juli*    {\bf K}_{\rm Redi} = \frac{1}{1 + |{\bf S}|^2}
7337b40cfd Navi*    \begin{pmatrix}
8679f9097b Jeff*    1 + S_y^2& -S_x S_y & S_x \\
                    -S_x S_y  & 1 + S_x^2 & S_y \\
7337b40cfd Navi*    S_x & S_y & |{\bf S}|^2 \\
                    \end{pmatrix}
8679f9097b Jeff* 
5b172de0d2 Jean* Here, :math:`S_x = \partial_x \sigma / (- \partial_z \sigma)`,
                 :math:`S_y = \partial_y \sigma / (- \partial_z \sigma)` are the components
                 of the isoneutral slope, and :math:`|{\bf S}|^2 = S_x^2 + S_y^2`.
8679f9097b Jeff* 
                 The first point to note is that a typical slope in the ocean interior is
                 small, say of the order :math:`10^{-4}`. A maximum slope might be of
                 order :math:`10^{-2}` and only exceeds such in unstratified regions
7337b40cfd Navi* where the slope is ill-defined. It is, therefore, justifiable, and
                 customary, to make the small-slope approximation, i.e., :math:`|{\bf S}| \ll 1`. Then
                 Redi projection tensor then simplifies to:
8679f9097b Jeff* 
                 .. math::
7337b40cfd Navi*    {\bf K}_{\rm Redi} =
                    \begin{pmatrix}
8679f9097b Jeff*    1 & 0 & S_x \\
                    0 & 1 & S_y \\
7337b40cfd Navi*    S_x & S_y & |{\bf S}|^2 \\
                    \end{pmatrix}
8679f9097b Jeff* 
a4576c7cde Juli* .. _GM_bolus_desc:
9c8516d9da Jeff* 
8679f9097b Jeff* GM parameterization
7337b40cfd Navi* -------------------
8679f9097b Jeff* 
7337b40cfd Navi*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 106.


 The GM parameterization aims to represent the advective or “transport”
8679f9097b Jeff*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 107.


 effect of geostrophic eddies by means of a “bolus” velocity,
7337b40cfd Navi* :math:`{\bf u}^\star`. The divergence of this advective flux is added to
8679f9097b Jeff* the tracer tendency equation (on the rhs):
                 
7337b40cfd Navi* .. math:: - \nabla \cdot ( \tau {\bf u}^\star )
8679f9097b Jeff* 
7337b40cfd Navi* The bolus velocity :math:`{\bf u}^\star` is defined as the rotational part
                 of a streamfunction
                 :math:`{\bf F}^\star = (F_x^\star, F_y^\star, 0)`:
8679f9097b Jeff* 
                 .. math::
                 
7337b40cfd Navi*    {\bf u}^\star = \nabla \times {\bf F}^\star =
                    \begin{pmatrix}
8679f9097b Jeff*    - \partial_z  F_y^\star \\
                    + \partial_z  F_x^\star \\
                    \partial_x F_y^\star - \partial_y F_x^\star
7337b40cfd Navi*    \end{pmatrix}
8679f9097b Jeff* 
                 and thus is automatically non-divergent. In the GM parameterization, the
                 streamfunction is specified in terms of the isoneutral slopes
                 :math:`S_x` and :math:`S_y`:
                 
                 .. math::
                 
                    \begin{aligned}
7337b40cfd Navi*    F_x^\star & = -\kappa_{\rm GM} S_y\\
                    F_y^\star & =  \kappa_{\rm GM} S_x
                    \end{aligned}
8679f9097b Jeff* 
                 with boundary conditions :math:`F_x^\star=F_y^\star=0` on upper and
7337b40cfd Navi* lower boundaries. :math:`\kappa_{\rm GM}` is colloquially called the isopycnal "thickness diffusivity"
                 or the "GM diffusivity". The bolus transport in the GM
                 parameterization is thus given by:
8679f9097b Jeff* 
                 .. math::
                 
7337b40cfd Navi*    {\bf u}^\star =
                    \begin{pmatrix}
8679f9097b Jeff*    u^\star \\
                    v^\star \\
                    w^\star
7337b40cfd Navi*    \end{pmatrix} =
                    \begin{pmatrix}
0bad585a21 Navi*    - \partial_z (\kappa_{\rm GM} S_x) \\
                    - \partial_z (\kappa_{\rm GM} S_y) \\
7337b40cfd Navi*    \partial_x (\kappa_{\rm GM} S_x) + \partial_y (\kappa_{\rm GM} S_y)
                    \end{pmatrix}
8679f9097b Jeff* 
7337b40cfd Navi* This is the "advective form" of the GM parameterization as applied by Danabasoglu and McWilliams (1995) :cite:`danabasoglu:95`,
                 employed in the GFDL Modular Ocean Model (MOM) versions 1 and 2. To use the advective form in MITgcm, set
                 :varlink:`GM_AdvForm` ``=.TRUE.`` in ``data.gmredi``
                 (also requires ``#define`` :varlink:`GM_BOLUS_ADVEC` and :varlink:`GM_EXTRA_DIAGONAL`).
                 As implemented in the MITgcm code, :math:`{\bf u}^\star` is simply added to Eulerian :math:`\vec{\bf u}`
                 (i.e., MITgcm variables :varlink:`uVel`, :varlink:`vVel`, :varlink:`wVel`)
                 and passed to tracer advection subroutines (:numref:`advection_schemes`)
                 unless :varlink:`GM_AdvSeparate` ``=.TRUE.`` in ``data.gmredi``, in which case the bolus transport is computed separately.
8679f9097b Jeff* 
7337b40cfd Navi* Note that in MITgcm, the variables for the GM bolus
                 streamfunction :varlink:`GM_PsiX` and :varlink:`GM_PsiY` are defined:
8679f9097b Jeff* 
                 .. math::
5b172de0d2 Jean*    \begin{aligned}
7337b40cfd Navi*    \begin{pmatrix}
                    \sf{GM\_PsiX} \\
                    \sf{GM\_PsiY}
                    \end{pmatrix} =
                    \begin{pmatrix}
0bad585a21 Navi*    \kappa_{\rm GM} S_x \\
                    \kappa_{\rm GM} S_y
7337b40cfd Navi*    \end{pmatrix} =
                    \begin{pmatrix}
8679f9097b Jeff*    F_y^\star \\
                    -F_x^\star
7337b40cfd Navi*    \end{pmatrix}
5b172de0d2 Jean*    \end{aligned}
                    :label: GM_bolus_psi
8679f9097b Jeff* 
9c8516d9da Jeff* .. _sub_gmredi_skewflux:
                 
8679f9097b Jeff* Griffies Skew Flux
7337b40cfd Navi* ------------------
8679f9097b Jeff* 
7337b40cfd Navi* Griffies (1998) :cite:`gr:98` notes that the discretization of bolus velocities involves multiple
8679f9097b Jeff* layers of differencing and interpolation that potentially lead to noisy
                 fields and computational modes. He pointed out that the bolus flux can
                 be re-written in terms of a non-divergent flux and a skew-flux:
                 
                 .. math::
                 
                    \begin{aligned}
7337b40cfd Navi*    {\bf u}^\star \tau
a4576c7cde Juli*    & =
7337b40cfd Navi*    \begin{pmatrix}
0bad585a21 Navi*    - \partial_z ( \kappa_{\rm GM} S_x ) \tau \\
                    - \partial_z ( \kappa_{\rm GM} S_y ) \tau \\
7337b40cfd Navi*    \Big[ \partial_x (\kappa_{\rm GM} S_x) + \partial_y (\kappa_{\rm GM} S_y) \Big] \tau
                    \end{pmatrix}
8679f9097b Jeff*    \\
a4576c7cde Juli*    & =
7337b40cfd Navi*    \begin{pmatrix}
0bad585a21 Navi*    - \partial_z ( \kappa_{\rm GM} S_x \tau) \\
                    - \partial_z ( \kappa_{\rm GM} S_y \tau) \\
                    \partial_x ( \kappa_{\rm GM} S_x \tau) + \partial_y ( \kappa_{\rm GM} S_y \tau)
7337b40cfd Navi*    \end{pmatrix}
                    + \kappa_{\rm GM} \begin{pmatrix}
                      S_x \partial_z \tau \\
                      S_y \partial_z \tau \\
                    -  S_x \partial_x \tau - S_y \partial_y \tau
                    \end{pmatrix}
                    \end{aligned}
8679f9097b Jeff* 
                 The first vector is non-divergent and thus has no effect on the tracer
                 field and can be dropped. The remaining flux can be written:
                 
0bad585a21 Navi* .. math:: \bf{u}^\star \tau = - \kappa_{\rm GM} \bf{K}_{\rm GM} \bf{\nabla} \tau
8679f9097b Jeff* 
                 where
                 
                 .. math::
                 
7337b40cfd Navi*    {\bf K}_{\rm GM} =
                    \begin{pmatrix}
                     0  &  0  & -S_x \\
                     0  &  0  & -S_y \\
                    S_x & S_y &   0
                    \end{pmatrix}
8679f9097b Jeff* 
                 is an anti-symmetric tensor.
                 
                 This formulation of the GM parameterization involves fewer derivatives
                 than the original and also involves only terms that already appear in
                 the Redi mixing scheme. Indeed, a somewhat fortunate cancellation
                 becomes apparent when we use the GM parameterization in conjunction with
                 the Redi isoneutral mixing scheme:
                 
                 .. math::
                 
7337b40cfd Navi*    \kappa_\rho {\bf K}_{\rm Redi} \nabla \tau
a4576c7cde Juli*    - {\bf u}^\star \tau =
7337b40cfd Navi*    ( \kappa_\rho {\bf K}_{\rm Redi} + \kappa_{\rm GM} {\bf K}_{\rm GM} ) \nabla \tau
8679f9097b Jeff* 
7337b40cfd Navi* If the Redi and GM diffusivities are equal, :math:`\kappa_{\rm GM} = \kappa_{\rho}`, then
8679f9097b Jeff* 
                 .. math::
7337b40cfd Navi*    \kappa_\rho {\bf K}_{\rm Redi} + \kappa_{\rm GM} {\bf K}_{\rm GM} =
8679f9097b Jeff*    \kappa_\rho
7337b40cfd Navi*    \begin{pmatrix}
8679f9097b Jeff*    1 & 0 & 0 \\
                    0 & 1 & 0 \\
a4576c7cde Juli*    2 S_x & 2 S_y & |{\bf S}|^2
7337b40cfd Navi*    \end{pmatrix}
8679f9097b Jeff* 
7337b40cfd Navi* which only differs from the variable Laplacian diffusion tensor by the two
8679f9097b Jeff* non-zero elements in the :math:`z`-row.
                 
                 .. admonition:: Subroutine
                   :class: note
                 
7337b40cfd Navi*   S/R GMREDI_CALC_TENSOR (:filelink:`pkg/gmredi/gmredi_calc_tensor.F`)
8679f9097b Jeff* 
5b172de0d2 Jean*   :math:`\sigma_x`: **sigmaX** (argument on entry)
                 
                   :math:`\sigma_y`: **sigmaY** (argument on entry)
                 
                   :math:`\sigma_z`: **sigmaR** (argument on entry)
                 
                 .. _sub_gmredi_in_P:
8679f9097b Jeff* 
5b172de0d2 Jean* Redi and GM schemes in pressure coordinate
                 ------------------------------------------
8679f9097b Jeff* 
5b172de0d2 Jean* When using pressure as a vertical coordinate (see :numref:`isomorphic-equations`),
                 the Redi scheme can be applied in the same way as in :math:`z-`\coordinates,
                 considering the slope of isoneutral surface relative to the model isobaric
                 surface, to rotate the diffusion operator along isoneutral surfaces.
                 
                 The two components of the slope relative to :math:`p-`\ coordinates are
                 :math:`S_x^p = \partial_x \sigma / \partial_p \sigma`,
                 :math:`S_y^p = \partial_y \sigma / \partial_p \sigma`.
                 Note that for convienience and consistency with the current implementation,
                 the sign of the slope in :math:`z-` or :math:`p-` coordinates is kept unchanged, i.e.,
                 identical to the sign of horizontal density gradient.
                 The negative sign is added back in the Redi tensor expression:
                 
                 .. math::
                    {\bf K}_{\rm Redi}^p =
                    \begin{pmatrix}
                       1   &    0   & -S_x^p \\
                       0   &    1   & -S_y^p \\
                    -S_x^p & -S_y^p & |{\bf S^p}|^2 \\
                    \end{pmatrix}
8679f9097b Jeff* 
5b172de0d2 Jean* In contrast, the GM scheme should instead consider the slope of isoneutral
                 surface relative to geopotential surface (constant :math:`z-`\ surface), so that its
                 effect will decrease the available potential energy and flatten the isopycnal.
                 Since :math:`dp = -(\rho g) dz`, the slope to consider would be, in two dimensions:
8679f9097b Jeff* 
5b172de0d2 Jean* .. math:: S_x^z = \partial_x \sigma / (- \partial_z \sigma) = \frac{1}{\rho g} S_x^p
                 
                 The effect on tracer :math:`\tau` from the bolus transport (:math:`{\bf u}^\star`)
                 advection would be:
                 
                 .. math:: [ {\bf u}^\star \cdot \nabla \tau ]^z
                    & = u^\star \partial_x \tau + w^\star \partial_z \tau
                    \\
                    & = \rho g \partial_p (\kappa_{\rm GM} \frac{1}{\rho g} S_x^p) \partial_x \tau
                      - \partial_x (\kappa_{\rm GM} \frac{1}{\rho g} S_x^p) (\rho g) \partial_p \tau
                    \\
                    & =  {\bf u}^{\star p} \cdot \nabla^p \tau
                 
                 .. math::
                    {\rm with:}~~
                    {\bf u}^{\star p} =
                    \begin{pmatrix}
                    u^{\star p} \\
                    v^{\star p} \\
                    \omega^{\star p}
                    \end{pmatrix} =
                    \begin{pmatrix}
                      \rho g \partial_p (\kappa_{\rm GM} \frac{1}{\rho g} S_x^p) \\
                      \rho g \partial_p (\kappa_{\rm GM} \frac{1}{\rho g} S_y^p) \\
                    - \rho g \partial_x (\kappa_{\rm GM} \frac{1}{\rho g} S_x^p)
                    - \rho g \partial_y (\kappa_{\rm GM} \frac{1}{\rho g} S_y^p)
                    \end{pmatrix}
                 .. :label: GM_bolus_in_p (note to me: this is commented out)
                 
                 This formulation above has not been implemented yet and instead only a simplified
                 version is available that ignores the difference between isobaric surfaces and
                 horizontal surfaces, as if in the above expression, the density :math:`\rho` were
                 uniform. This approximation seems valid for the ocean where the
                 isopycnal slope is generally much larger than the isobaric slope relative to
                 horizontal surface.
                 
                 With this approxmation, the expression of the bolus transport simplifies
                 and becomes "isomorphic" to the :math:`z-`\ coordinate form :eq:`GM_bolus_psi`,
                 with the sign reversal of all three components of the bolus transport,
                 due to the expression of the curl in a left-handed coordinate system.
8679f9097b Jeff* 
a4576c7cde Juli* .. _sub_gmredi_visbeck:
                 
7337b40cfd Navi* Visbeck et al. 1997 GM diffusivity :math:`\kappa_{GM}(x,y)`
                 -----------------------------------------------------------
8679f9097b Jeff* 
7337b40cfd Navi* Visbeck et al. (1997) :cite:`visbeck:97` suggest making the eddy coefficient,
                 :math:`\kappa_{\rm GM}`, a function of
                 the Eady growth rate, :math:`|f|/\sqrt{\rm Ri}`. The formula involves a
8679f9097b Jeff* non-dimensional constant, :math:`\alpha`, and a length-scale :math:`L`:
                 
7337b40cfd Navi* .. math:: \kappa_{\rm GM} = \alpha L^2 \overline{ \frac{|f|}{\sqrt{\rm Ri}} }^z
8679f9097b Jeff* 
                 where the Eady growth rate has been depth averaged (indicated by the
                 over-line). A local Richardson number is defined
7337b40cfd Navi* :math:`{\rm Ri} = N^2 / (\partial_z u)^2` which, when combined with thermal wind gives:
8679f9097b Jeff* 
                 .. math::
                 
7337b40cfd Navi*    \frac{1}{\rm Ri} = \frac{(\partial u/\partial z)^2}{N^2} =
                    \frac{ \left ( \dfrac{g}{f \rho_0} | \nabla \sigma | \right )^2 }{N^2} =
8679f9097b Jeff*    \frac{ M^4 }{ |f|^2 N^2 }
                 
7337b40cfd Navi* where :math:`M^2 = g | \nabla \sigma| / \rho_0`. Substituting into
0bad585a21 Navi* the formula for :math:`\kappa_{\rm GM}` gives:
8679f9097b Jeff* 
                 .. math::
                 
0bad585a21 Navi*    \kappa_{\rm GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z =
8679f9097b Jeff*    \alpha L^2 \overline{ \frac{M^2}{N^2} N }^z =
7337b40cfd Navi*    \alpha L^2 \overline{ |{\bf S}| N }^z
8679f9097b Jeff* 
a4576c7cde Juli* Marshall et al. 2012 GM diffusivity :math:`\kappa_{GM}(x,y)`
                 ------------------------------------------------------------
                 
                 Marshall et al. (2012) :cite:`marshall:12b` via the GEOMETRIC framework suggest
                 that the eddy coefficient :math:`\kappa_{\rm GM}` be a linear function of a
                 parameterized total eddy energy :math:`E`, a non-dimensional constant
                 :math:`\alpha` (namelist parameter :varlink:`GEOM_alpha`; note
                 bounded in magnitude by 1 in the QG setting) and the
                 model stratification. A key impact of GEOMETRIC is to make the sensitivity of
                 the Antarctic Circumpolar Current (ACC) and AMOC to changes in the Southern
                 Ocean wind forcing closer to those in analogous high resolution models (e.g.,
                 Mak et al. 2018 :cite:`mak:18`). Following Mak et al. (2022) :cite:`mak:22`, the
                 depth-integrated coefficient is given by
                 
                 .. math::
                    \kappa_{\rm GM} = \alpha \frac{\int E\; \mathrm{d}z}{\int \Gamma (M^2 / N)\; \mathrm{d}z} \Gamma(z) = \alpha \frac{\int E\; \mathrm{d}z}{\int \Gamma |{\bf S}| N\; \mathrm{d}z} \Gamma(z),
                 
                 where notation is as :numref:`sub_gmredi_visbeck`, noting that :math:`|{\bf S}|
                 = M^2 / N^2` is the magnitude of the isopycnal slopes. :math:`\Gamma(z)` is some
                 vertical structure function that can be prescribed if required; default is
                 :math:`\Gamma(z)\equiv1`, with an option for :math:`\Gamma(z)\equiv N^2 /
                 N^2_{\rm ref}` following Ferreria et al. (2005) :cite:`ferriera:05`,
                 activated by setting parameter :varlink:`GEOM_vert_struc` ``= .TRUE.``. The
                 depth-integrated eddy energy follows its own prognostic equation as
                 
                 .. math::
                    \frac{\mathrm{d}}{\mathrm{d} t} \int E\; \mathrm{d}z +
                    \nabla \cdot \left( (\tilde{\boldsymbol{u}} - |c|\boldsymbol{e}_x ) \int E\; \mathrm{d}z \right) =
                    \int \kappa_{\rm gm} |{\bf S}|^2 N^2\; \mathrm{d}z -
                    \lambda \int E\; \mathrm{d}z +
                    \nu_E \nabla^2 \int E\; \mathrm{d}z
                 
                 Terms above, respectively, are the time-evolution, advection, source, dissipation and diffusion of
                 depth-integrated parameterized total eddy energy, with
                 :math:`\tilde{\boldsymbol{u}}` the depth-averaged velocity, :math:`c` a
                 long Rossby phase speed (computed from a WKB approximation, rotated accordingly if ``usingCurvilinearGrid = .TRUE.``), :math:`\lambda` a
                 dissipation rate (namelist parameter :varlink:`GEOM_lmbda`), and :math:`\nu_E` an energy
                 diffusion coefficient (namelist parameter :varlink:`GEOM_diffKh_EKE`). Note that bit-for-bit
                 restart reproducibility GEOM requires an additional pickup file (``pickup_gmredi``). 
                 
                 .. admonition:: note
                   :class: note
                 
                   The present GEOMETRIC implementation is strictly for the GM coefficient and
                   overrides any other specifications in ``data.gmredi`` for the GM coefficient.
                   The choice of Redi coefficient, clipping/tapering options and others however
                   are still controlled by the analogous entries in ``data.gmredi``.
                 
9c8516d9da Jeff* .. _sub_gmredi_tapering_stability:
                 
8679f9097b Jeff* Tapering and stability
7337b40cfd Navi* ----------------------
8679f9097b Jeff* 
                 Experience with the GFDL model showed that the GM scheme has to be
                 matched to the convective parameterization. This was originally
                 expressed in connection with the introduction of the KPP boundary layer
7337b40cfd Navi* scheme (Large et al. 1994 :cite:`lar-eta:94`) but in fact, as subsequent experience with the MIT model has
8679f9097b Jeff* found, is necessary for any convective parameterization.
                 
                 Slope clipping
                 ++++++++++++++
                 
                 Deep convection sites and the mixed layer are indicated by homogenized,
                 unstable or nearly unstable stratification. The slopes in such regions
                 can be either infinite, very large with a sign reversal or simply very
                 large. From a numerical point of view, large slopes lead to large
                 variations in the tensor elements (implying large bolus flow) and can be
7337b40cfd Navi* numerically unstable. This was first recognized by Cox (1987) :cite:`cox87` who implemented
8679f9097b Jeff*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 445.


 “slope clipping” in the isopycnal mixing tensor. Here, the slope
                 magnitude is simply restricted by an upper limit:
                 
                 .. math::
                 
                    \begin{aligned}
7337b40cfd Navi*    |\nabla_h \sigma| & = \sqrt{ \sigma_x^2 + \sigma_y^2 }\\
a4576c7cde Juli*    S_{\rm lim} & = - \frac{|\nabla_h \sigma|}{ S_{\max} },
7337b40cfd Navi*    \quad \mbox{where $S_{\max}>0$ is a parameter} \\
                    \sigma_z^\star & = \min( \sigma_z, S_{\rm lim} ) \\
                    {[s_x, s_y]} & = - \frac{ [\sigma_x, \sigma_y] }{\sigma_z^\star}
                    \end{aligned}
8679f9097b Jeff* 
                 Notice that this algorithm assumes stable stratification through the
7337b40cfd Navi*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 459.


 “min” function. In the case where the fluid is well stratified
0bad585a21 Navi* (:math:`\sigma_z < S_{\rm lim}`) then the slopes evaluate to:
8679f9097b Jeff* 
7337b40cfd Navi* .. math:: {[s_x, s_y]} = - \frac{ [\sigma_x, \sigma_y] }{\sigma_z}
8679f9097b Jeff* 
0bad585a21 Navi* while in the limited regions (:math:`\sigma_z > S_{\rm lim}`) the slopes
8679f9097b Jeff* become:
                 
7337b40cfd Navi* .. math:: {[s_x, s_y]} = \frac{ [\sigma_x, \sigma_y] }{|\nabla_h \sigma| / S_{\max}}
8679f9097b Jeff* 
                 so that the slope magnitude is limited :math:`\sqrt{s_x^2 + s_y^2} =
7337b40cfd Navi* S_{\max}`.
8679f9097b Jeff* 
                 The slope clipping scheme is activated in the model by setting
7337b40cfd Navi*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 473.


 :varlink:`GM_taper_scheme` ``= ’clipping’`` in ``data.gmredi``.
8679f9097b Jeff* 
                 Even using slope clipping, it is normally the case that the vertical
                 diffusion term (with coefficient :math:`\kappa_\rho{\bf K}_{33} =
7337b40cfd Navi* \kappa_\rho S_{\max}^2`) is large and must be time-stepped using an
                 implicit procedure (see :numref:`implicit-backward-stepping`). Fig.
8679f9097b Jeff* [fig-mixedlayer] shows the mixed layer depth resulting from a) using the
                 GM scheme with clipping and b) no GM scheme (horizontal diffusion). The
                 classic result of dramatically reduced mixed layers is evident. Indeed,
                 the deep convection sites to just one or two points each and are much
                 shallower than we might prefer. This, it turns out, is due to the over
                 zealous re-stratification due to the bolus transport parameterization.
                 Limiting the slopes also breaks the adiabatic nature of the GM/Redi
                 parameterization, re-introducing diabatic fluxes in regions where the
                 limiting is in effect.
                 
7337b40cfd Navi* .. admonition:: Subroutine
                   :class: note
                 
                   S/R GMREDI_SLOPE_LIMIT (:filelink:`pkg/gmredi/gmredi_slope_limit.F`)
8679f9097b Jeff* 
7337b40cfd Navi*   :math:`\sigma_x, s_x`: **SlopeX** (argument)
                 
                   :math:`\sigma_y, s_y`: **SlopeY** (argument)
                 
                   :math:`\sigma_z`: **dSigmadRReal** (argument)
8679f9097b Jeff* 
7337b40cfd Navi*   :math:`z_\sigma^{*}`: **dRdSigmaLtd** (argument)
                 
                 Tapering: Gerdes, Koberle and Willebrand, 1991 (GKW91)
                 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
                 
                 The tapering scheme used in Gerdes et al. (1991) :cite:`gkw:91` (GKW91)
                 addressed two issues with the clipping
8679f9097b Jeff* method: the introduction of large vertical fluxes in addition to
                 convective adjustment fluxes is avoided by tapering the GM/Redi slopes
                 back to zero in low-stratification regions; the adjustment of slopes is
                 replaced by a tapering of the entire GM/Redi tensor. This means the
                 direction of fluxes is unaffected as the amplitude is scaled.
                 
                 The scheme inserts a tapering function, :math:`f_1(S)`, in front of the
                 GM/Redi tensor:
                 
7337b40cfd Navi* .. math:: f_1(S) = \min \left[ 1, \left( \frac{S_{\max}}{|{\bf S}|}\right)^2 \right]
8679f9097b Jeff* 
7337b40cfd Navi* where :math:`S_{\max}` is the maximum slope you want allowed. Where the
                 slopes, :math:`|{\bf S}|<S_{\max}` then :math:`f_1(S) = 1` and the tensor is
                 un-tapered but where :math:`|{\bf S}| \ge S_{\max}` then :math:`f_1(S)` scales
8679f9097b Jeff* down the tensor so that the effective vertical diffusivity term
7337b40cfd Navi* :math:`\kappa f_1(S) |{\bf S}|^2 = \kappa S_{\max}^2`.
8679f9097b Jeff* 
                 The GKW91 tapering scheme is activated in the model by setting
7337b40cfd Navi*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 525.


 :varlink:`GM_taper_scheme` ``= ’gkw91’`` in ``data.gmredi``.
                 
                 .. figure:: figs/tapers.*
                     :width: 70%
                     :align: center
                     :alt: Tapering for GM scheme
                     :name: tapers
                 
a4576c7cde Juli*     Taper functions used in GKW91 and DM95.
7337b40cfd Navi* 
                 .. figure:: figs/effective_slopes.*
                     :width: 70%
                     :align: center
                     :alt: Tapering for GM scheme
                     :name: effective_slopes
8679f9097b Jeff* 
7337b40cfd Navi*     Effective slope as a function of 'true' slope using Cox slope clipping, GKW91 limiting and DM95 limiting.
8679f9097b Jeff* 
7337b40cfd Navi* Tapering: Danabasoglu and McWilliams, 1995 (DM95)
                 +++++++++++++++++++++++++++++++++++++++++++++++++
8679f9097b Jeff* 
7337b40cfd Navi* The tapering scheme used by Danabasoglu and McWilliams (1995) :cite:`danabasoglu:95` (DM95)
                 followed a similar procedure but used a different tapering function, :math:`f_1(S)`:
8679f9097b Jeff* 
7337b40cfd Navi* .. math:: f_1(S) = \frac{1}{2} \left[ 1+\tanh \left( \frac{S_c - |{\bf S}|}{S_d} \right) \right]
8679f9097b Jeff* 
                 where :math:`S_c = 0.004` is a cut-off slope and :math:`S_d=0.001` is a
                 scale over which the slopes are smoothly tapered. Functionally, the
                 operates in the same way as the GKW91 scheme but has a substantially
7337b40cfd Navi* lower cut-off, turning off the GM/Redi parameterization for weaker
8679f9097b Jeff* slopes.
                 
                 The DM95 tapering scheme is activated in the model by setting
7337b40cfd Navi*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 558.


 :varlink:`GM_taper_scheme` ``= ’dm95’`` in ``data.gmredi``.
8679f9097b Jeff* 
7337b40cfd Navi* Tapering: Large, Danabasoglu and Doney, 1997 (LDD97)
                 ++++++++++++++++++++++++++++++++++++++++++++++++++++
8679f9097b Jeff* 
7337b40cfd Navi* The tapering used in Large et al. (1997) :cite:`lar-eta:97` (LDD97)
                 is based on the DM95 tapering scheme, but also
8679f9097b Jeff* tapers the scheme with an additional function of height, :math:`f_2(z)`,
7337b40cfd Navi* so that the GM/Redi subgrid-scale fluxes are reduced near the surface:
8679f9097b Jeff* 
7337b40cfd Navi* .. math:: f_2(z) = \frac{1}{2} \left[ 1 + \sin \left(\pi \frac{z}{D} - \frac{\pi}{2} \right) \right]
8679f9097b Jeff* 
7337b40cfd Navi* where :math:`D = (c / f) |{\bf S}|` is a depth scale, with :math:`f` the
                 Coriolis parameter and :math:`c=2`�m/s (corresponding to the first baroclinic wave speed, so that :math:`c/f` is the Rossby radius).
                 This tapering that varies with depth
                 was introduced to fix some spurious interaction with the mixed-layer KPP
8679f9097b Jeff* parameterization.
                 
                 The LDD97 tapering scheme is activated in the model by setting
7337b40cfd Navi*
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 577.


 :varlink:`GM_taper_scheme` ``= ’ldd97’`` in ``data.gmredi``.
                 
                 .. _ssub_phys_pkg_gmredi_config:
                 
                 GMREDI configuration and compiling
                 ==================================
                 
                 Compile-time options
                 --------------------
                 
                 As with all MITgcm packages, GMREDI can be turned on or off at compile time
                 (see :numref:`building_code`)
                 
                 - using the ``packages.conf`` file by adding ``gmredi`` to it
                 
                 - or using :filelink:`genmake2 <tools/genmake2>` adding ``-enable=gmredi`` or
                   ``-disable=gmredi`` switches
                 
                 - **required packages and CPP options**:
                   :filelink:`gmredi <pkg/gmredi>` requires
                 
                 Parts of the :filelink:`gmredi <pkg/gmredi>` code can be enabled or disabled at
                 compile time via CPP preprocessor flags. These options are set in
                 :filelink:`GMREDI_OPTIONS.h <pkg/gmredi/GMREDI_OPTIONS.h>`.
                 :numref:`tab_phys_pkg_gmredi_cpp` summarizes the most important ones. For additional
                 options see :filelink:`GMREDI_OPTIONS.h <pkg/gmredi/GMREDI_OPTIONS.h>`.
                 
                 .. tabularcolumns:: |\Y{.375}|\Y{.1}|\Y{.55}|
                 
                 .. csv-table:: Some of the most relevant CPP preprocessor flags in the :filelink:`gmredi <pkg/gmredi>` package.
                    :header: "CPP option", "Default", "Description"
                    :widths: 30, 10, 60
                    :name: tab_phys_pkg_gmredi_cpp
                 
                    :varlink:`GM_NON_UNITY_DIAGONAL`, #define, allows the leading diagonal (top two rows) to be non-unity
                    :varlink:`GM_EXTRA_DIAGONAL`, #define, allows different values of :math:`\kappa_{\rm GM}` and :math:`\kappa_{\rho}`; also required for advective form
                    :varlink:`GM_BOLUS_ADVEC`, #define, allows use of the advective form (bolus velocity)
                    :varlink:`GM_BOLUS_BVP`, #define, allows use of Boundary-Value-Problem method to evaluate bolus transport
                    :varlink:`ALLOW_GM_LEITH_QG`, #undef, allow QG Leith variable viscosity to be added to GMRedi coefficient
                    :varlink:`GM_VISBECK_VARIABLE_K`, #undef, allows Visbeck et al. formulation to compute :math:`\kappa_{\rm GM}`
a4576c7cde Juli*    :varlink:`GM_GEOM_VARIABLE_K`, #undef, allows Marshall et al. formulation to compute :math:`\kappa_{\rm GM}`
7337b40cfd Navi* 
                 .. _ssub_phys_pkg_gmredi_runtime:
                 
                 Run-time parameters
                 ===================
                 
                 Run-time parameters (see :numref:`tab_phys_pkg_gmredi_runtimeparms`) are set in
                 ``data.gmredi`` (read in :filelink:`pkg/gmredi/gmredi_readparms.F`).
                 
                 Enabling the package
                 --------------------
                 
                 :filelink:`gmredi <pkg/gmredi>` package is switched on/off at run-time by
                 setting :varlink:`useGMREDI` ``= .TRUE.,`` in ``data.pkg``.
                 
                 General flags and parameters
                 ----------------------------
                 
                 :numref:`tab_phys_pkg_gmredi_runtimeparms` lists most run-time parameters.
                 
                 .. tabularcolumns:: |\Y{.275}|\Y{.20}|\Y{.525}|
                 
                 .. table:: Run-time parameters and default values
                   :class: longtable
                   :name: tab_phys_pkg_gmredi_runtimeparms
                 
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   |   Name                             |      Default value           |   Description                                                           |
                   +====================================+==============================+=========================================================================+
                   | :varlink:`GM_AdvForm`              |     FALSE                    | use advective form (bolus velocity); FALSE uses skewflux form           |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_AdvSeparate`          |     FALSE                    | do advection by Eulerian and bolus velocity separately                  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_background_K`         |     0.0                      | thickness diffusivity :math:`\kappa_{\rm GM}` (m\ :sup:`2`\ /s)         |
                   |                                    |                              | (GM bolus transport)                                                    |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_isopycK`              |   :varlink:`GM_background_K` | isopycnal diffusivity :math:`\kappa_{\rho}` (m\ :sup:`2`\ /s)           |
                   |                                    |                              | (Redi tensor)                                                           |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_maxSlope`             |     1.0E-02                  | maximum slope (tapering/clipping)                                       |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Kmin_horiz`           |     0.0                      | minimum horizontal diffusivity (m\ :sup:`2`\ /s)                        |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Small_Number`         |     1.0E-20                  | :math:`\epsilon` used in computing the slope                            |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_slopeSqCutoff`        |     1.0E+48                  | :math:`|{\bf S}|^2` cut-off value for zero taper function               |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_taper_scheme`         |     ' '                      | taper scheme option ('orig', 'clipping', 'fm07', 'stableGmAdjTap',      |
                   |                                    |                              | 'linear', 'ac02', 'gkw91', 'dm95', 'ldd97')                             |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_maxTransLay`          |     500.0                    | maximum transition layer thickness (m)                                  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_facTrL2ML`            |     5.0                      | maximum trans. layer thick. as a factor of local mixed-layer depth      |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_facTrL2dz`            |     1.0                      | minimum trans. layer thick. as a factor of local dr                     |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Scrit`                |     0.004                    | :math:`S_c` parameter for 'dm95' and 'ldd97 ' tapering function         |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Sd`                   |     0.001                    | :math:`S_d` parameter for 'dm95' and 'ldd97' tapering function          |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_UseBVP`               |     FALSE                    | use Boundary-Value-Problem method for bolus transport                   |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_BVP_ModeNumber`       |     1                        | vertical mode number used for speed :math:`c` in BVP transport          |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_BVP_cMin`             |     1.0E-01                  | minimum value for wave speed parameter :math:`c` in BVP (m/s)           |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_UseSubMeso`           |     FALSE                    | use sub-mesoscale eddy parameterization for bolus transport             |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`subMeso_Ceff`            |     7.0E-02                  | efficiency coefficient of mixed-layer eddies                            |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`subMeso_invTau`          |     2.0E-06                  | inverse of mixing timescale in sub-meso parameterization (s\ :sup:`-1`) |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`subMeso_LfMin`           |     1.0E+03                  | minimum value for length-scale :math:`L_f` (m)                          |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`subMeso_Lmax`            |     110.0E+03                | maximum horizontal grid-scale length (m)                                |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Visbeck_alpha`        |     0.0                      | :math:`\alpha` parameter for Visbeck et al. scheme (non-dim.)           |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Visbeck_length`       |     200.0E+03                | :math:`L` length scale parameter for Visbeck et al. scheme (m)          |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Visbeck_depth`        |     1000.0                   | depth (m) over which to average in computing Visbeck                    |
                   |                                    |                              | :math:`\kappa_{\rm GM}`                                                 |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Visbeck_maxSlope`     |     :varlink:`GM_maxSlope`   | maximum slope used in computing Visbeck et al. :math:`\kappa_{\rm GM}`  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Visbeck_minVal_K`     |     0.0                      | minimum :math:`\kappa_{\rm GM}` (m\ :sup:`2`\ /s) using Visbeck et al.  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_Visbeck_maxVal_K`     |     2500.0                   | maximum :math:`\kappa_{\rm GM}` (m\ :sup:`2`\ /s) using Visbeck et al.  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
c54b782cac rosi*   | :varlink:`GM_useGEOM`              |     FALSE                    | use GEOM scheme if #define :varlink:`GM_GEOM_VARIABLE_K`                |
a4576c7cde Juli*   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_alpha`              |     0.06                     | :math:`\alpha` parameter for GEOM scheme (non-dim.)                     |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_lmbda`              |     1.16E-07                 | eddy energy dissipation rate for GEOM scheme (s\ :sup:`-1`)             |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_diffKh_EKE`         |     5.0E+02                  | eddy energy diffusion coefficient for GEOM scheme (m\ :sup:`2`\ /s)     |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_ini_EKE`            |     1.0E-03                  | initial value for parameterized total depth eddy energy for GEOM scheme |
                   |                                    |                              | (m\ :sup:`3`\ / s\ :sup:`2`)                                            |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_vert_struc`         |     FALSE                    | use imposed vertical structure function in GEOM scheme                  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_vert_struc_min`     |     0.1                      | minimum value of vertical structure function in GEOM scheme (non-dim.)  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_vert_struc_max`     |     1.0                      | maximum value of vertical structure function in GEOM scheme (non-dim.)  |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_minVal_K`           |     0.0                      | minimum :math:`\kappa_{\rm GM}` (m\ :sup:`2`\ /s) using GEOM scheme     |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GEOM_maxVal_K`           |     2500.0                   | maximum :math:`\kappa_{\rm GM}` (m\ :sup:`2`\ /s) using GEOM scheme     |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
7337b40cfd Navi*   | :varlink:`GM_useLeithQG`           |     FALSE                    | add Leith QG viscosity to GMRedi tensor                                 |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_iso2dFile`            |     ' '                      | input file for 2D (:math:`x,y`) scaling of isopycnal diffusivity        |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_iso1dFile`            |     ' '                      | input file for 1D vert. scaling of isopycnal diffusivity                |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_bol2dFile`            |     ' '                      | input file for 2D (:math:`x,y`) scaling of thickness diffusivity        |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_bol1dFile`            |     ' '                      | input file for 1D vert. scaling of thickness diffusivity                |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_background_K3dFile`   |     ' '                      | input file for 3D (:math:`x,y,r`) :varlink:`GM_background_K`            |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_isopycK3dFile`        |     ' '                      | input file for 3D (:math:`x,y,r`) :varlink:`GM_isopycK`                 |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
                   | :varlink:`GM_MNC`                  |     :varlink:`useMNC`        | write GMREDI snapshot output using :filelink:`/pkg/mnc`                 |
                   +------------------------------------+------------------------------+-------------------------------------------------------------------------+
8679f9097b Jeff* 
                 .. _ssub_phys_pkg_gmredi_diagnostics:
                 
7337b40cfd Navi* GMREDI Diagnostics
                 ==================
8679f9097b Jeff* 
                 ::
                 
7337b40cfd Navi*    ----------------------------------------------------------------------
                    <-Name->|Levs|<- code ->|<--  Units   -->|<- Description
                    ----------------------------------------------------------------------
                    GM_VisbK|  1 |SM P    M1|m^2/s           |Mixing coefficient from Visbeck etal parameterization
                    GM_hTrsL|  1 |SM P    M1|m               |Base depth (>0) of the Transition Layer
                    GM_baseS|  1 |SM P    M1|1               |Slope at the base of the Transition Layer
                    GM_rLamb|  1 |SM P    M1|1/m             |Slope vertical gradient at Trans. Layer Base (=recip.Lambda)
                    SubMesLf|  1 |SM P    M1|m               |Sub-Meso horiz. Length Scale (Lf)
                    SubMpsiX|  1 |UU      M1|m^2/s           |Sub-Meso transp.stream-funct. magnitude (Psi0): U component
                    SubMpsiY|  1 |VV      M1|m^2/s           |Sub-Meso transp.stream-funct. magnitude (Psi0): V component
                    GM_Kux  | 18 |UU P    MR|m^2/s           |K_11 element (U.point, X.dir) of GM-Redi tensor
                    GM_Kvy  | 18 |VV P    MR|m^2/s           |K_22 element (V.point, Y.dir) of GM-Redi tensor
                    GM_Kuz  | 18 |UU      MR|m^2/s           |K_13 element (U.point, Z.dir) of GM-Redi tensor
                    GM_Kvz  | 18 |VV      MR|m^2/s           |K_23 element (V.point, Z.dir) of GM-Redi tensor
                    GM_Kwx  | 18 |UM      LR|m^2/s           |K_31 element (W.point, X.dir) of GM-Redi tensor
                    GM_Kwy  | 18 |VM      LR|m^2/s           |K_32 element (W.point, Y.dir) of GM-Redi tensor
                    GM_Kwz  | 18 |WM P    LR|m^2/s           |K_33 element (W.point, Z.dir) of GM-Redi tensor
                    GM_PsiX | 18 |UU      LR|m^2/s           |GM Bolus transport stream-function : U component
                    GM_PsiY | 18 |VV      LR|m^2/s           |GM Bolus transport stream-function : V component
                    GM_KuzTz| 18 |UU      MR|degC.m^3/s      |Redi Off-diagonal Temperature flux: X component
                    GM_KvzTz| 18 |VV      MR|degC.m^3/s      |Redi Off-diagonal Temperature flux: Y component
                    GM_KwzTz| 18 |WM      LR|degC.m^3/s      |Redi main-diagonal vertical Temperature flux
                    GM_ubT  | 18 |UUr     MR|degC.m^3/s      |Zonal Mass-Weight Bolus Transp of Pot Temp
                    GM_vbT  | 18 |VVr     MR|degC.m^3/s      |Meridional Mass-Weight Bolus Transp of Pot Temp
                    GM_BVPcW|  1 |SU P    M1|m/s             |WKB wave speed (at Western edge location)
                    GM_BVPcS|  1 |SV P    M1|m/s             |WKB wave speed (at Southern edge location)
a4576c7cde Juli*    GM_GEOMK| 15 |SMRP    MR|m/s^2           |GEOM 3d kgm field
                    GEOMeE  |  1 |SM P    M1|m^3/s^2         |GEOM parameterized depth-int eddy energy
                    GEOMstru| 15 |SMRP    MR|                |spatial structure function
                    GEOMEgen|  1 |SM P    M1|m^3/s^3         |GEOM eddy energy generation tendency
                    GEOMEdis|  1 |SM P    M1|m^3/s^3         |GEOM eddy energy dissipation tendency
                    GEOMEadv|  1 |SM P    M1|m^3/s^3         |GEOM eddy energy advective tendency
                    GEOMEwav|  1 |SM P    M1|m^3/s^3         |GEOM eddy energy wave advection tendency
                    GEOMElap|  1 |SM P    M1|m^3/s^3         |GEOM eddy energy diffusion tendency
                    GEOM_c1 |  1 |SM P    M1|m/s             |first baroclinic wave phase speed
                    GEOMcros|  1 |SM P    M1|m/s             |signed long Rossby wave phase speed
7337b40cfd Navi* 
                 Experiments and tutorials that use GMREDI
                 =========================================
8679f9097b Jeff* 
0bad585a21 Navi* -  Southern Ocean Reentrant Channel Example, in :filelink:`verification/tutorial_reentrant_channel`,
                    described in :numref:`sec_eg_reentrant_channel`
                 
                 -  Global Ocean Simulation, in :filelink:`verification/tutorial_global_oce_latlon`,
                    described in :numref:`sec_global_oce_latlon`
8679f9097b Jeff* 
7337b40cfd Navi* -  Front Relax experiment, in :filelink:`verification/front_relax`
8679f9097b Jeff* 
7337b40cfd Navi* -  Ideal 2D Ocean experiment, in :filelink:`verification/ideal_2D_oce`.

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