ROMS/TOMS Developers

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 (m²/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.