A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. This occurs when horizontal mixing is required on tracer or momentum (ln_traldf_hor or ln_dynldf_hor) in - or mixed -- coordinates, and isoneutral mixing is required whatever the vertical coordinate is. This direction of mixing is defined by its slopes in the i- and j-directions at the face of the cell of the quantity to be diffused. For a tracer, this leads to the following four slopes : , , , (see (5.9)), while for momentum the slopes are , , , for and , , , for .
In -coordinates, geopotential mixing ( horizontal mixing) and are the slopes between the geopotential and computational surfaces. Their discrete formulation is found by locally solving (5.9) when the diffusive fluxes in the three directions are set to zero and is assumed to be horizontally uniform, a linear function of , the depth of a -point.
These slopes are computed once in ldfslp_init when ln_sco=True, and either ln_traldf_hor=True or ln_dynldf_hor=True.
As the mixing is performed along neutral surfaces, the gradient of in (9.11) has to be evaluated at the same local pressure (which, in decibars, is approximated by the depth in meters in the model). Therefore (9.11) cannot be used as such, but further transformation is needed depending on the vertical coordinate used:
Note: The solution for -coordinate passes trough the use of different (and better) expression for the constraint on iso-neutral fluxes. Following Griffies , instead of specifying directly that there is a zero neutral diffusive flux of locally referenced potential density, we stay in the - plane and consider the balance between the neutral direction diffusive fluxes of potential temperature and salinity:
This constraint leads to the following definition for the slopes:
Note that such a formulation could be also used in the -coordinate and -coordinate with partial steps cases.
This implementation is a rather old one. It is similar to the one proposed by Cox , except for the background horizontal diffusion. Indeed, the Cox implementation of isopycnal diffusion in GFDL-type models requires a minimum background horizontal diffusion for numerical stability reasons. To overcome this problem, several techniques have been proposed in which the numerical schemes of the ocean model are modified [Weaver and Eby, 1997, Griffies et al., 1998]. Griffies's scheme is now available in NEMO if traldf_grif_iso is set true; see Appdx D. Here, another strategy is presented [Lazar, 1997]: a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of grid point noise generated by the iso-neutral diffusion operator (Fig. 9.1). This allows an iso-neutral diffusion scheme without additional background horizontal mixing. This technique can be viewed as a diffusion operator that acts along large-scale (2 x) iso-neutral surfaces. The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of an horizontal background mixing.
Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, contrary to the Griffies et al.  operator which has that property.
For numerical stability reasons [Griffies, 2004, Cox, 1987], the slopes must also be bounded by everywhere. This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to at metres and thereafter decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface).
yellowadd here a discussion about the flattening of the slopes, vs tapering the coefficient.
The iso-neutral diffusion operator on momentum is the same as the one used on tracers but applied to each component of the velocity separately (see (6.27) in section 6.6.2). The slopes between the surface along which the diffusion operator acts and the surface of computation (- or -surfaces) are defined at -, -, and uw- points for the -component, and -, - and vw- points for the -component. They are computed from the slopes used for tracer diffusion, (9.10) and (9.11) :
The major issue remaining is in the specification of the boundary conditions. The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces, i.e. using the shear computed along the model levels and with no additional friction at the ocean bottom (see §8.1).
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17