- The Grid
- The Depth Mean Solutions
- Advection
- Horizontal Pressure Gradients
- Vertical Mixing
- Convective Adjustment
- The Initial and Boundary Conditions

The Grid

The water column at each grid point is divided into (typically
the order of 20 or 30) -levels and Fig. 1
shows a schematic of the levels at a grid
point. These are indexed , with levels and
lying respectively below the sea bed and above the sea surface to
facilitate the use of flux boundary conditions, hence levels
are interior to the model. To maintain resolution near the surface in
deep water we allow the spacing of -levels to vary in the
horizontal, according to the transformation

(27) |

where are evenly spaced levels between and . The deviation from the usual -levels is given by

(28) |

The equations for the surface elevation (Eqn. 14) and depth mean currents (Eqns. 4 and 5) are integrated forward in time (using centred space forward time differencing) with a barotropic time step which is a fraction of the (long) baroclinic time step . The surface elevations are filtered using the method described by Killworth et al. (1991) to prevent the "checkerboard" pattern of grid-scale noise which is sometimes found when using the B-grid.

Advection

In order to maintain horizontal gradients and minimise numerical
diffusion the advection of momentum and scalars uses the `Piecewise
Parabolic Method' (Colella and Woodward 1984, James 1996). This scheme
assumes the variables vary parabolically across the grid boxes, with
the mean in the grid box, , taken to be the value at the
grid-point, . The parabolas
are then defined by

(29) |

Horizontal Pressure Gradients

(30) |

(31) |

for , and

(32) |

The pressure gradients along the edges of the plane are then

(33) |

(34) |

This technique allows a straightforward treatment of cases where the sea bed lies above the plane at one or more of the surrounding -points; if either of the corners on an edge are below the sea bed then the corresponding pressure gradient takes the last defined value above it. This assumes the isopycnals are horizontal close to the sea bed and does not reflect the correct boundary condition for temperature and salinity (zero normal gradients). However, the diffusive layers in which the thermoclines and haloclines curve to meet this condition are not well resolved in this model.

This method has been tested against the pressure gradient due to an analytically defined thermocline and gives significantly more accurate results than the conventional method when the thermocline is flat or sloping in the opposite sense to the -levels. Both methods give equally good results when the thermocline slopes in the same sense as the -levels.

Instead of the turbulence closure scheme we use a simple convective adjustment procedure to mix unstable density gradients more effectively. At each horizontal grid point the model searches down in the vertical for a point where , at -level . The temperature and salinity at is then repeatedly averaged with points below until the resulting buoyancy (here an overbar indicates an average from to ). The temperature and salinity in this interval is then replaced by and . If necessary this procedure is repeated for points above . The velocity field is also adjusted by convection; prior to the adjustment at b-points, the and values are interpolated onto the u-points and are averaged in the vertical according to the above scheme.

Because of its arbitrary nature, the use of this convective adjustment scheme is less than satisfactory. This is particularly the case for velocities; while its effects on their profiles are not great it may result in excessive mixing under unstratified conditions. However, we do find the use of this adjustment somewhat improves the comparison with SST observations, but a more satisfactory method of treating statically unstable conditions needs to be sought.

Temperature and salinity may be relaxed to climatological values in a
region next to the model boundaries. These values, ,
are taken from the boundary data.
We find these temperature and salinity fields often contain some
small scale variations, for example on the continental slope, which can be
removed
by smoothing
on horizontal planes since these variations cannot adjust
and may drive erroneous currents.
The effects of this smoothing are examined in Holt et al. (2001).
For a relaxation zone 4 points wide,
the temperature at the grid points from the western boundary are given by

(35) |

The sea surface is forced by data from a weather prediction model and the temperature updated by the heat fluxes calculated from bulk formulae using this data following Gill (1982) and downwelling solar radiation (with a decay scale taken to be 0.154m). Details of these are given by Holt and James (1999a) and will not be repeated here. At present there is no surface salinity flux in this model; however fresh water inputs are determined from daily discharge data from rivers plus, for applications including the North Sea, the exchange with the Baltic and the Kattegat.