8.4.1. GMREDI: Gent-McWilliams/Redi Eddy Parameterization Introduction

Package 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 [Red82]), 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 [GM90], Gent et al. 1995 [GWMM95]), 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 Section CPP options enable or disable different aspects of the package (Section Run-time options, flags, and filenames are set in data.gmredi (Section Available diagnostics output is listed in Section Description

The first GCM implementation of the Redi scheme was by Cox (1987) [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) [GGP+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. Redi scheme: Isopycnal diffusion

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:

\[\begin{split}{\bf K}_{\rm Redi} = \frac{1}{1 + |{\bf S}|^2} \begin{pmatrix} 1 + S_y^2& -S_x S_y & S_x \\ -S_x S_y & 1 + S_x^2 & S_y \\ S_x & S_y & |{\bf S}|^2 \\ \end{pmatrix}\end{split}\]

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:

\[\begin{split}{\bf K}_{\rm Redi} = \begin{pmatrix} 1 & 0 & S_x \\ 0 & 1 & S_y \\ S_x & S_y & |{\bf S}|^2 \\ \end{pmatrix}\end{split}\] GM parameterization

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 \({\bf F}^\star = (F_x^\star, F_y^\star, 0)\):

\[\begin{split}{\bf u}^\star = \nabla \times {\bf F}^\star = \begin{pmatrix} - \partial_z F_y^\star \\ + \partial_z F_x^\star \\ \partial_x F_y^\star - \partial_y F_x^\star \end{pmatrix}\end{split}\]

and thus is automatically non-divergent. In the GM parameterization, the streamfunction is specified in terms of the isoneutral slopes \(S_x\) and \(S_y\):

\[\begin{split}\begin{aligned} F_x^\star & = -\kappa_{\rm GM} S_y\\ F_y^\star & = \kappa_{\rm GM} S_x \end{aligned}\end{split}\]

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:

\[\begin{split}{\bf u}^\star = \begin{pmatrix} u^\star \\ v^\star \\ w^\star \end{pmatrix} = \begin{pmatrix} - \partial_z (\kappa_{\rm GM} S_x) \\ - \partial_z (\kappa_{\rm GM} S_y) \\ \partial_x (\kappa_{\rm GM} S_x) + \partial_y (\kappa_{\rm GM} S_y) \end{pmatrix}\end{split}\]

This is the “advective form” of the GM parameterization as applied by Danabasoglu and McWilliams (1995) [DJCM95], employed in the GFDL Modular Ocean Model (MOM) versions 1 and 2. To use the advective form in MITgcm, set GM_AdvForm =.TRUE. in data.gmredi (also requires #define GM_BOLUS_ADVEC and GM_EXTRA_DIAGONAL). As implemented in the MITgcm code, \({\bf u}^\star\) is simply added to Eulerian \(\vec{\bf u}\) (i.e., MITgcm variables uVel, vVel, wVel) and passed to tracer advection subroutines (Section 2.17) unless 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 GM_PsiX and GM_PsiY are defined:

\[\begin{split}\begin{pmatrix} \sf{GM\_PsiX} \\ \sf{GM\_PsiY} \end{pmatrix} = \begin{pmatrix} \kappa_{\rm GM} S_x \\ \kappa_{\rm GM} S_y \end{pmatrix} = \begin{pmatrix} F_y^\star \\ -F_x^\star \end{pmatrix}\end{split}\] Griffies Skew Flux

Griffies (1998) [Gri98] 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:

\[\begin{split}\begin{aligned} {\bf u}^\star \tau & = \begin{pmatrix} - \partial_z ( \kappa_{\rm GM} S_x ) \tau \\ - \partial_z ( \kappa_{\rm GM} S_y ) \tau \\ \Big[ \partial_x (\kappa_{\rm GM} S_x) + \partial_y (\kappa_{\rm GM} S_y) \Big] \tau \end{pmatrix} \\ & = \begin{pmatrix} - \partial_z ( \kappa_{\rm GM} S_x \tau) \\ - \partial_z ( \kappa_{\rm GM} S_y \tau) \\ \partial_x ( \kappa_{\rm GM} S_x \tau) + \partial_y ( \kappa_{\rm GM} S_y \tau) \end{pmatrix} + \kappa_{\rm GM} \begin{pmatrix} S_x \partial_z \tau \\ S_y \partial_z \tau \\ - S_x \partial_x \tau - S_y \partial_y \tau \end{pmatrix} \end{aligned}\end{split}\]

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:

\[\bf{u}^\star \tau = - \kappa_{\rm GM} \bf{K}_{\rm GM} \bf{\nabla} \tau\]


\[\begin{split}{\bf K}_{\rm GM} = \begin{pmatrix} 0 & 0 & -S_x \\ 0 & 0 & -S_y \\ S_x & S_y & 0 \end{pmatrix}\end{split}\]

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

\[\begin{split}\kappa_\rho {\bf K}_{\rm Redi} + \kappa_{\rm GM} {\bf K}_{\rm GM} = \kappa_\rho \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 S_x & 2 S_y & |{\bf S}|^2 \end{pmatrix}\end{split}\]

which only differs from the variable Laplacian diffusion tensor by the two non-zero elements in the \(z\)-row.


S/R GMREDI_CALC_TENSOR (pkg/gmredi/gmredi_calc_tensor.F)

\(\sigma_x\): SlopeX (argument on entry)

\(\sigma_y\): SlopeY (argument on entry)

\(\sigma_z\): SlopeY (argument)

\(S_x\): SlopeX (argument on exit)

\(S_y\): SlopeY (argument on exit) Visbeck et al. 1997 GM diffusivity \(\kappa_{GM}(x,y)\)

Visbeck et al. (1997) [VMHS97] 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:

\[\kappa_{\rm GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z = \alpha L^2 \overline{ \frac{M^2}{N^2} N }^z = \alpha L^2 \overline{ |{\bf S}| N }^z\] Tapering and stability

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 [LMD94]) but in fact, as subsequent experience with the MIT model has 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 numerically unstable. This was first recognized by Cox (1987) [Cox87] who implemented “slope clipping” in the isopycnal mixing tensor. Here, the slope magnitude is simply restricted by an upper limit:

\[\begin{split}\begin{aligned} |\nabla_h \sigma| & = \sqrt{ \sigma_x^2 + \sigma_y^2 }\\ S_{\rm lim} & = - \frac{|\nabla_h \sigma|}{ S_{\max} }, \quad \mbox{where $S_{\max}>0$ is a parameter} \\ \sigma_z^\star & = \min( \sigma_z, S_{\rm lim} ) \\ {[s_x, s_y]} & = - \frac{ [\sigma_x, \sigma_y] }{\sigma_z^\star} \end{aligned}\end{split}\]

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 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 Section 2.6). 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.


S/R GMREDI_SLOPE_LIMIT (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)

Tapering: Gerdes, Koberle and Willebrand, 1991 (GKW91)

The tapering scheme used in Gerdes et al. (1991) [GKW91] (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 GM_taper_scheme = ’gkw91’ in data.gmredi.

Tapering for GM scheme

Figure 8.7 Taper functions used in GKW91 and DM95.

Tapering for GM scheme

Figure 8.8 Effective slope as a function of ‘true’ slope using Cox slope clipping, GKW91 limiting and DM95 limiting.

Tapering: Danabasoglu and McWilliams, 1995 (DM95)

The tapering scheme used by Danabasoglu and McWilliams (1995) [DJCM95] (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 GM_taper_scheme = ’dm95’ in data.gmredi.

Tapering: Large, Danabasoglu and Doney, 1997 (LDD97)

The tapering used in Large et al. (1997) [LDDM97] (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 GM_taper_scheme = ’ldd97’ in data.gmredi. GMREDI configuration and compiling Compile-time options

As with all MITgcm packages, GMREDI can be turned on or off at compile time (see Section 3.5)

  • using the packages.conf file by adding gmredi to it

  • or using genmake2 adding -enable=gmredi or -disable=gmredi switches

  • required packages and CPP options: gmredi requires

Parts of the gmredi code can be enabled or disabled at compile time via CPP preprocessor flags. These options are set in GMREDI_OPTIONS.h. Table 8.5 summarizes the most important ones. For additional options see GMREDI_OPTIONS.h.

Table 8.5 Some of the most relevant CPP preprocessor flags in the gmredi package.

CPP option





allows the leading diagonal (top two rows) to be non-unity



allows different values of \(\kappa_{\rm GM}\) and \(\kappa_{\rho}\); also required for advective form



allows use of the advective form (bolus velocity)



allows use of Boundary-Value-Problem method to evaluate bolus transport



allow QG Leith variable viscosity to be added to GMRedi coefficient



allows Visbeck et al. formulation to compute \(\kappa_{\rm GM}\) Run-time parameters

Run-time parameters (see Table 8.6) are set in data.gmredi (read in pkg/gmredi/gmredi_readparms.F). Enabling the package

gmredi package is switched on/off at run-time by setting useGMREDI = .TRUE., in data.pkg. General flags and parameters

Table 8.6 lists most run-time parameters.

Table 8.6 Run-time parameters and default values


Default value




use advective form (bolus velocity); FALSE uses skewflux form



do advection by Eulerian and bolus velocity separately



thickness diffusivity \(\kappa_{\rm GM}\) (m2/s) (GM bolus transport)



isopycnal diffusivity \(\kappa_{\rho}\) (m2/s) (Redi tensor)



maximum slope (tapering/clipping)



minimum horizontal diffusivity (m2/s)



\(\epsilon\) used in computing the slope



\(|{\bf S}|^2\) cut-off value for zero taper function


‘ ‘

taper scheme option (‘orig’, ‘clipping’, ‘fm07’, ‘stableGmAdjTap’, ‘linear’, ‘ac02’, ‘gkw91’, ‘dm95’, ‘ldd97’)



maximum transition layer thickness (m)



maximum trans. layer thick. as a factor of local mixed-layer depth



minimum trans. layer thick. as a factor of local dr



\(S_c\) parameter for ‘dm95’ and ‘ldd97 ‘ tapering function



\(S_d\) parameter for ‘dm95’ and ‘ldd97’ tapering function



use Boundary-Value-Problem method for bolus transport



vertical mode number used for speed \(c\) in BVP transport



minimum value for wave speed parameter \(c\) in BVP (m/s)



use sub-mesoscale eddy parameterization for bolus transport



efficiency coefficient of mixed-layer eddies



inverse of mixing timescale in sub-meso parameterization (s-1)



minimum value for length-scale \(L_f\) (m)



maximum horizontal grid-scale length (m)



\(\alpha\) parameter for Visbeck et al. scheme (non-dim.)



\(L\) length scale parameter for Visbeck et al. scheme (m)



depth (m) over which to average in computing Visbeck \(\kappa_{\rm GM}\)



maximum slope used in computing Visbeck et al. \(\kappa_{\rm GM}\)



minimum \(\kappa_{\rm GM}\) (m2/s) using Visbeck et al.



maximum \(\kappa_{\rm GM}\) (m2/s) using Visbeck et al.



add Leith QG viscosity to GMRedi tensor


‘ ‘

input file for 2D (\(x,y\)) scaling of isopycnal diffusivity


‘ ‘

input file for 1D vert. scaling of isopycnal diffusivity


‘ ‘

input file for 2D (\(x,y\)) scaling of thickness diffusivity


‘ ‘

input file for 1D vert. scaling of thickness diffusivity


‘ ‘

input file for 3D (\(x,y,r\)) GM_background_K


‘ ‘

input file for 3D (\(x,y,r\)) GM_isopycK



write GMREDI snapshot output using /pkg/mnc GMREDI Diagnostics

<-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) Experiments and tutorials that use GMREDI