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 isentropes (neutral surfaces) by means of a diffusion operator oriented along the local isentropic surface. The second part, GM (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 is a function of slope of the isentropic surfaces.
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`.
The first GCM implementation of the Redi scheme was by Cox (1987) :cite:`cox87` in the GFDL ocean circulation model. The original approach failed to distinguish between isopycnals and surfaces of locally referenced potential density (now called neutral surfaces), which are proper isentropes for the ocean. As 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 non-divergent bolus velocity. The method defines two streamfunctions 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. 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 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.
The Redi scheme diffuses tracers along isopycnals and introduces a term in the tendency (rhs) of such a tracer (here \tau) of the form:
\nabla \cdot ( \kappa_\rho {\bf K}_{\rm Redi} \nabla \tau )
where \kappa_\rho is the along isopycnal diffusivity and {\bf K}_{\rm Redi} is a rank 2 tensor that projects the gradient of \tau onto the isopycnal surface. The unapproximated projection tensor is:
{\bf K}_{\rm Redi} = \frac{1}{1 + |{\bf S}|^2}
\begin{pmatrix}
1 + S_y^2& -S_x S_y & S_x \\
-S_x S_y & 1 + S_x^2 & S_y \\
S_x & S_y & |{\bf S}|^2 \\
\end{pmatrix}
Here, S_x = \partial_x \sigma / (- \partial_z \sigma), S_y = \partial_y \sigma / (- \partial_z \sigma) are the components of the isoneutral slope, and |{\bf S}|^2 = S_x^2 + S_y^2.
The first point to note is that a typical slope in the ocean interior is small, say of the order 10^{-4}. A maximum slope might be of order 10^{-2} and only exceeds such in unstratified regions where the slope is ill-defined. It is, therefore, justifiable, and customary, to make the small-slope approximation, i.e., |{\bf S}| \ll 1. Then Redi projection tensor then simplifies to:
{\bf K}_{\rm Redi} =
\begin{pmatrix}
1 & 0 & S_x \\
0 & 1 & S_y \\
S_x & S_y & |{\bf S}|^2 \\
\end{pmatrix}
The GM parameterization aims to represent the advective or “transport” effect of geostrophic eddies by means of a “bolus” velocity, {\bf u}^\star. The divergence of this advective flux is added to the tracer tendency equation (on the rhs):
- \nabla \cdot ( \tau {\bf u}^\star )
The bolus velocity {\bf u}^\star is defined as the rotational part of a streamfunction {\bf F}^\star = (F_x^\star, F_y^\star, 0):
{\bf u}^\star = \nabla \times {\bf F}^\star =
\begin{pmatrix}
- \partial_z F_y^\star \\
+ \partial_z F_x^\star \\
\partial_x F_y^\star - \partial_y F_x^\star
\end{pmatrix}
and thus is automatically non-divergent. In the GM parameterization, the streamfunction is specified in terms of the isoneutral slopes S_x and S_y:
\begin{aligned}
F_x^\star & = -\kappa_{\rm GM} S_y\\
F_y^\star & = \kappa_{\rm GM} S_x
\end{aligned}
with boundary conditions F_x^\star=F_y^\star=0 on upper and lower boundaries. \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:
{\bf u}^\star =
\begin{pmatrix}
u^\star \\
v^\star \\
w^\star
\end{pmatrix} =
\begin{pmatrix}
- \partial_z (\kappa_{\rm GM} S_x) \\
- \partial_z (\kappa_{\rm GM} S_y) \\
\partial_x (\kappa_{\rm GM} S_x) + \partial_y (\kappa_{\rm GM} S_y)
\end{pmatrix}
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, {\bf u}^\star is simply added to Eulerian \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.
Note that in MITgcm, the variables for the GM bolus streamfunction :varlink:`GM_PsiX` and :varlink:`GM_PsiY` are defined:
\begin{aligned}
\begin{pmatrix}
\sf{GM\_PsiX} \\
\sf{GM\_PsiY}
\end{pmatrix} =
\begin{pmatrix}
\kappa_{\rm GM} S_x \\
\kappa_{\rm GM} S_y
\end{pmatrix} =
\begin{pmatrix}
F_y^\star \\
-F_x^\star
\end{pmatrix}
\end{aligned}
Griffies (1998) :cite:`gr:98` notes that the discretization of bolus velocities involves multiple 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:
\begin{aligned}
{\bf u}^\star \tau
& =
\begin{pmatrix}
- \partial_z ( \kappa_{\rm GM} S_x ) \tau \\
- \partial_z ( \kappa_{\rm GM} S_y ) \tau \\
\Big[ \partial_x (\kappa_{\rm GM} S_x) + \partial_y (\kappa_{\rm GM} S_y) \Big] \tau
\end{pmatrix}
\\
& =
\begin{pmatrix}
- \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)
\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}
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:
\bf{u}^\star \tau = - \kappa_{\rm GM} \bf{K}_{\rm GM} \bf{\nabla} \tau
where
{\bf K}_{\rm GM} =
\begin{pmatrix}
0 & 0 & -S_x \\
0 & 0 & -S_y \\
S_x & S_y & 0
\end{pmatrix}
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:
\kappa_\rho {\bf K}_{\rm Redi} \nabla \tau
- {\bf u}^\star \tau =
( \kappa_\rho {\bf K}_{\rm Redi} + \kappa_{\rm GM} {\bf K}_{\rm GM} ) \nabla \tau
If the Redi and GM diffusivities are equal, \kappa_{\rm GM} = \kappa_{\rho}, then
\kappa_\rho {\bf K}_{\rm Redi} + \kappa_{\rm GM} {\bf K}_{\rm GM} =
\kappa_\rho
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
2 S_x & 2 S_y & |{\bf S}|^2
\end{pmatrix}
which only differs from the variable Laplacian diffusion tensor by the two non-zero elements in the z-row.
Subroutine
S/R GMREDI_CALC_TENSOR (:filelink:`pkg/gmredi/gmredi_calc_tensor.F`)
\sigma_x: sigmaX (argument on entry)
\sigma_y: sigmaY (argument on entry)
\sigma_z: sigmaR (argument on entry)
When using pressure as a vertical coordinate (see :numref:`isomorphic-equations`), the Redi scheme can be applied in the same way as in 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 p-coordinates are S_x^p = \partial_x \sigma / \partial_p \sigma, 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 z- or 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:
{\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}
In contrast, the GM scheme should instead consider the slope of isoneutral surface relative to geopotential surface (constant z-surface), so that its effect will decrease the available potential energy and flatten the isopycnal. Since dp = -(\rho g) dz, the slope to consider would be, in two dimensions:
S_x^z = \partial_x \sigma / (- \partial_z \sigma) = \frac{1}{\rho g} S_x^p
The effect on tracer \tau from the bolus transport ({\bf u}^\star) advection would be:
[ {\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
{\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}
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 \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 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.
Visbeck et al. (1997) :cite:`visbeck:97` suggest making the eddy coefficient, \kappa_{\rm GM}, a function of the Eady growth rate, |f|/\sqrt{\rm Ri}. The formula involves a non-dimensional constant, \alpha, and a length-scale L:
\kappa_{\rm GM} = \alpha L^2 \overline{ \frac{|f|}{\sqrt{\rm Ri}} }^z
where the Eady growth rate has been depth averaged (indicated by the over-line). A local Richardson number is defined {\rm Ri} = N^2 / (\partial_z u)^2 which, when combined with thermal wind gives:
\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} =
\frac{ M^4 }{ |f|^2 N^2 }
where M^2 = g | \nabla \sigma| / \rho_0. Substituting into the formula for \kappa_{\rm GM} gives:
\kappa_{\rm GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z =
\alpha L^2 \overline{ \frac{M^2}{N^2} N }^z =
\alpha L^2 \overline{ |{\bf S}| N }^z
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 scheme (Large et al. 1994 :cite:`lar-eta:94`) but in fact, as subsequent experience with the MIT model has found, is necessary for any convective parameterization.
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 numerically unstable. This was first recognized by Cox (1987) :cite:`cox87` who implemented “slope clipping” in the isopycnal mixing tensor. Here, the slope magnitude is simply restricted by an upper limit:
\begin{aligned}
|\nabla_h \sigma| & = \sqrt{ \sigma_x^2 + \sigma_y^2 }\\
S_{\rm lim} & = - \frac{|\nabla_h \sigma|}{ S_{\max} },
\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}
Notice that this algorithm assumes stable stratification through the “min” function. In the case where the fluid is well stratified (\sigma_z < S_{\rm lim}) then the slopes evaluate to:
{[s_x, s_y]} = - \frac{ [\sigma_x, \sigma_y] }{\sigma_z}
while in the limited regions (\sigma_z > S_{\rm lim}) the slopes become:
{[s_x, s_y]} = \frac{ [\sigma_x, \sigma_y] }{|\nabla_h \sigma| / S_{\max}}
so that the slope magnitude is limited \sqrt{s_x^2 + s_y^2} = S_{\max}.
The slope clipping scheme is activated in the model by setting
:varlink:`GM_taper_scheme` = ’clipping’ in data.gmredi.
Even using slope clipping, it is normally the case that the vertical diffusion term (with coefficient \kappa_\rho{\bf K}_{33} = \kappa_\rho S_{\max}^2) is large and must be time-stepped using an implicit procedure (see :numref:`implicit-backward-stepping`). Fig. [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.
Subroutine
S/R GMREDI_SLOPE_LIMIT (:filelink:`pkg/gmredi/gmredi_slope_limit.F`)
\sigma_x, s_x: SlopeX (argument)
\sigma_y, s_y: SlopeY (argument)
\sigma_z: dSigmadRReal (argument)
z_\sigma^{*}: dRdSigmaLtd (argument)
The tapering scheme used in Gerdes et al. (1991) :cite:`gkw:91` (GKW91) addressed two issues with the clipping 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, f_1(S), in front of the GM/Redi tensor:
f_1(S) = \min \left[ 1, \left( \frac{S_{\max}}{|{\bf S}|}\right)^2 \right]
where S_{\max} is the maximum slope you want allowed. Where the slopes, |{\bf S}|<S_{\max} then f_1(S) = 1 and the tensor is un-tapered but where |{\bf S}| \ge S_{\max} then f_1(S) scales down the tensor so that the effective vertical diffusivity term \kappa f_1(S) |{\bf S}|^2 = \kappa S_{\max}^2.
The GKW91 tapering scheme is activated in the model by setting
:varlink:`GM_taper_scheme` = ’gkw91’ in data.gmredi.
Taper functions used in GKW91 and DM95.
Effective slope as a function of 'true' slope using Cox slope clipping, GKW91 limiting and DM95 limiting.
The tapering scheme used by Danabasoglu and McWilliams (1995) :cite:`danabasoglu:95` (DM95) followed a similar procedure but used a different tapering function, f_1(S):
f_1(S) = \frac{1}{2} \left[ 1+\tanh \left( \frac{S_c - |{\bf S}|}{S_d} \right) \right]
where S_c = 0.004 is a cut-off slope and 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 lower cut-off, turning off the GM/Redi parameterization for weaker slopes.
The DM95 tapering scheme is activated in the model by setting
:varlink:`GM_taper_scheme` = ’dm95’ in data.gmredi.
The tapering used in Large et al. (1997) :cite:`lar-eta:97` (LDD97) is based on the DM95 tapering scheme, but also tapers the scheme with an additional function of height, f_2(z), so that the GM/Redi subgrid-scale fluxes are reduced near the surface:
f_2(z) = \frac{1}{2} \left[ 1 + \sin \left(\pi \frac{z}{D} - \frac{\pi}{2} \right) \right]
where D = (c / f) |{\bf S}| is a depth scale, with f the Coriolis parameter and c=2 m/s (corresponding to the first baroclinic wave speed, so that c/f is the Rossby radius). This tapering that varies with depth was introduced to fix some spurious interaction with the mixed-layer KPP parameterization.
The LDD97 tapering scheme is activated in the model by setting
:varlink:`GM_taper_scheme` = ’ldd97’ in data.gmredi.
As with all MITgcm packages, GMREDI can be turned on or off at compile time (see :numref:`building_code`)
- using the
packages.conffile by addinggmredito it - or using :filelink:`genmake2 <tools/genmake2>` adding
-enable=gmredior-disable=gmrediswitches - 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}|
Some of the most relevant CPP preprocessor flags in the :filelink:`gmredi <pkg/gmredi>` package.
| CPP option | Default | Description |
|---|---|---|
| :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 \kappa_{\rm GM} and \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 \kappa_{\rm GM} |
Run-time parameters (see :numref:`tab_phys_pkg_gmredi_runtimeparms`) are set in
data.gmredi (read in :filelink:`pkg/gmredi/gmredi_readparms.F`).
:filelink:`gmredi <pkg/gmredi>` package is switched on/off at run-time by
setting :varlink:`useGMREDI` = .TRUE., in data.pkg.
:numref:`tab_phys_pkg_gmredi_runtimeparms` lists most run-time parameters.
.. tabularcolumns:: |\Y{.275}|\Y{.20}|\Y{.525}|
Run-time parameters and default values
| 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 \kappa_{\rm GM} (m2/s) (GM bolus transport) |
| :varlink:`GM_isopycK` | :varlink:`GM_background_K` | isopycnal diffusivity \kappa_{\rho} (m2/s) (Redi tensor) |
| :varlink:`GM_maxSlope` | 1.0E-02 | maximum slope (tapering/clipping) |
| :varlink:`GM_Kmin_horiz` | 0.0 | minimum horizontal diffusivity (m2/s) |
| :varlink:`GM_Small_Number` | 1.0E-20 | \epsilon used in computing the slope |
| :varlink:`GM_slopeSqCutoff` | 1.0E+48 | |{\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 | S_c parameter for 'dm95' and 'ldd97 ' tapering function |
| :varlink:`GM_Sd` | 0.001 | 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 c in BVP transport |
| :varlink:`GM_BVP_cMin` | 1.0E-01 | minimum value for wave speed parameter 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-1) |
| :varlink:`subMeso_LfMin` | 1.0E+03 | minimum value for length-scale L_f (m) |
| :varlink:`subMeso_Lmax` | 110.0E+03 | maximum horizontal grid-scale length (m) |
| :varlink:`GM_Visbeck_alpha` | 0.0 | \alpha parameter for Visbeck et al. scheme (non-dim.) |
| :varlink:`GM_Visbeck_length` | 200.0E+03 | 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 \kappa_{\rm GM} |
| :varlink:`GM_Visbeck_maxSlope` | :varlink:`GM_maxSlope` | maximum slope used in computing Visbeck et al. \kappa_{\rm GM} |
| :varlink:`GM_Visbeck_minVal_K` | 0.0 | minimum \kappa_{\rm GM} (m2/s) using Visbeck et al. |
| :varlink:`GM_Visbeck_maxVal_K` | 2500.0 | maximum \kappa_{\rm GM} (m2/s) using Visbeck et al. |
| :varlink:`GM_useLeithQG` | FALSE | add Leith QG viscosity to GMRedi tensor |
| :varlink:`GM_iso2dFile` | ' ' | input file for 2D (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 (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 (x,y,r) :varlink:`GM_background_K` |
| :varlink:`GM_isopycK3dFile` | ' ' | input file for 3D (x,y,r) :varlink:`GM_isopycK` |
| :varlink:`GM_MNC` | :varlink:`useMNC` | write GMREDI snapshot output using :filelink:`/pkg/mnc` |
---------------------------------------------------------------------- <-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)
- 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`
- Front Relax experiment, in :filelink:`verification/front_relax`
- Ideal 2D Ocean experiment, in :filelink:`verification/ideal_2D_oce`.