Following Weaver and Courtier (2001), the vertical and horizontal correlations, C, for any state variable Φ are modeled using separated 1D and 2D diffusion equations:

where K_{v} and K_{H} are the vertical and horizontal diffusion coefficients (m^{2}/s), assumed to be of order one. They are set to unity (isotropic correlations), if these coefficients are not found in ROMS/TOMS background/model standard deviation NetCDF file (STDname, variables K_{v} and K_{h}). The user can set K_{v} and K_{h} to the desired patterns to get anisotropic correlations. If we discretize the above diffusion operators using an explicit FTCS scheme, the stability limit is governed by:

In practice, we set γ < 1 to be below the stability threshold, (γ = 0.5). This stability parameter is set in 4DVAR input script. There are two variables: Vgamma and Hgamma for vertical and horizontal diffusion operators. In realistic applications, the vertical difussion needs to be solved implicitly (IMPLICIT_VCONV). Otherwise, the it will be very expensive since the vertical scales are much smaller. The implicit algorithm is unconditionally stable for any time-step, so we use the maximum thickness in the above equation instead of the minimum.

The parameters controlling the vertical and horizontal length-scales of the Gaussian correlation function are:

over a time interval 0 ≤ t ≤ T and

where M_{v} and M_{H} are the total number of vertical and horizontal integration steps. Notice that both values are forced to an even number since the squared-root diffusion operators are used to insure symmetry. The squared-root operator iterates only for half of the steps. For any given decorrelation length-scale, the number of integration steps are given by:

The vertical and horizontal decorrelation scales (m) are set in 4DVar input script variables Vdecay(:) and Hdecay(:) for each state variable.

ROMS/TOMS reports to standard output all the parameters associated with these diffusion operators. For example,

5.0000E-01 Hgamma Horizontal diffusion stability/accuracy factor. 5.0000E-05 Vgamma Vertical diffusion stability/accuracy factor. 20000.000 Hdecay(isFsur) Horizontal decorrelation scale (m), free-sruface. 20000.000 Hdecay(isUbar) Horizontal decorrelation scale (m), 2D U-momentum. 20000.000 Hdecay(isVbar) Horizontal decorrelation scale (m), 2D V-momentum. 20000.000 Hdecay(isUvel) Horizontal decorrelation scale (m), 3D U-momentum. 20000.000 Hdecay(isVvel) Horizontal decorrelation scale (m), 3D V-momentum. 20000.000 Hdecay(idTvar) Horizontal decorrelation scale (m), temp. 20000.000 Hdecay(idTvar) Horizontal decorrelation scale (m), salt. 5.000 Vdecay(isUvel) Vertical decorrelation scale (m), 3D U-momentum. 5.000 Vdecay(isVvel) Vertical decorrelation scale (m), 3D V-momentum. 5.000 Vdecay(idTvar) Vertical decorrelation scale (m), temp. 5.000 Vdecay(idTvar) Vertical decorrelation scale (m), salt. Minimum X-grid spacing, DXmin = 5.26835784E+00 km Maximum X-grid spacing, DXmax = 5.58390264E+00 km Minimum Y-grid spacing, DYmin = 5.05224079E+00 km Maximum Y-grid spacing, DYmax = 5.35460591E+00 km Minimum Z-grid spacing, DZmin = 1.33333333E-01 m Maximum Z-grid spacing, DZmax = 2.86550363E+02 m Minimum barotropic Courant Number = 2.99123135E-02 Maximum barotropic Courant Number = 8.17259140E-01 Maximum Coriolis Courant Number = 3.43845829E-02 Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s zeta Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s ubar Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s vbar Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s u Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s v Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s temp Horizontal convolution, NHsteps, DTsizeH = 64 3.19064213E+06 s salt Vertical convolution, NVsteps, DTsizeV = 6 2.05277776E+00 s u Vertical convolution, NVsteps, DTsizeV = 6 2.05277776E+00 s v Vertical convolution, NVsteps, DTsizeV = 6 2.05277776E+00 s temp Vertical convolution, NVsteps, DTsizeV = 6 2.05277776E+00 s salt

It is recommended to reduce the Vgamma parameter until a value of M_{v} is around six for vertical diffusion accuracy.