- Introduction
- Model Description: The Model Equations.
- The Equation of State
- The Equations of Motion.
- Turbulence Closure

This section of the POLCOMS documentation is a general description of the model, including the model equations, the model forcing and the numerical methods employed. Much of this material has been published in Holt and James (2001). It ends with a bibliography including published work using POLCOMS.

The physical model used in POLCOMS is based on the Proudman Oceanographic Laboratory three-dimensional baroclinic model POL3DB and has been developed to incorporate features suitable for the modelling of baroclinic processes on the shelf, at the shelf-slope and in ocean regions to allow long term coupled ocean-shelf simulations. Versions of this model have been running operationally at the UK Met Office since 2000, though its development can be traced back to the mid 1980s, as shown in the bibliography.

The simulation of seasonal currents, water quality parameters and plankton variability in shelf seas requires a physical model which goes beyond the well established tide and surge models to include the effects of horizontal and vertical density variations. This is necessary since the former introduces seasonal transports not seen in constant density models, while the latter controls the vertical fluxes crucial to biological production. Moreover thermal fronts affect both the horizontal and vertical transport of tracers. To this end a three-dimensional model with temperature and salinity treated as prognostic variables has been developed: the Proudman Oceanographic Laboratory Three-dimensional Baroclinic B-grid model (POL3DB). The origins of this model lie with James (1986) and it has subsequently been developed (James 1996) to include a sophisticated advection scheme, the `Piecewise Parabolic Method' (PPM) described below. This has excellent feature-preserving properties making it ideal for the simulation of on-shelf baroclinic features such as river plumes (James 1997) and fronts (Proctor and James 1996), and the transport of tracers from localised sources (Holt and James 1999b). Moreover, the model is formulated on an Arakawa B-grid (see section 3.1), in contrast to the C-grid used in many shelf sea models, for example the Princeton Ocean Model (Blumberg and Mellor 1987). The B-grid is more commonly used in deep ocean models, for example Killworth et al. (1991) and is well suited to the modelling of horizontal density variations since the Coriolis term can be calculated without averaging. This helps prevent the dispersion of the velocity features associated with fronts, in contrast with the C-grid which requires averaging over a number of points to calculate this term. A disadvantage is that continuity and scalar advection require averaging of velocities not required on the C-grid. It is this choice of grid and the non-diffusive advection which particularly distinguish this model from others.

Since the model is designed to cover sea areas which include both shelf and deep-sea regions it includes a number of techniques pertinent to the treatment of deep water: although the model equations remain in -coordinates in the vertical, the spacing of these coordinate surfaces on the finite difference grid is allowed to vary in the horizontal according to the s-coordinate transform of Song and Haidvogel (1994) (see section 3.1); the pressure gradient calculations are made by interpolation onto horizontal planes through the points where velocities are defined (section 49.1); and a term for the variation of compressibility with temperature and salinity (section 2.3) is included in the model equation of state for sea water. Horizontal diffusion is included as an option, but it is not needed for model stability.

The Equation of State

The density is defined by an approximation to the full UNESCO equation
of state:
, where is the
potential temperature (C), the salinity (p.s.u.)
and the pressure relative
to the sea surface. is taken from the UNESCO equation of
state and

(1) |

(2) |

For accuracy in the numerical calculation we define the buoyancy
, where the 'potential' buoyancy is
(kgm is the reference
density)
and the variation of compressibility with temperature and salinity is
accounted for by
, with
.
Our initial condition gives
. The total
(hydrostatic) pressure is then given by

(3) |

and

while the equations for the depth varying components are:

(6) |

(7) |

(8) |

(9) |

(10) |

The advection terms are given by

(11) |

(12) |

(13) |

We use slip vertical boundary conditions: the components of
surface and bottom stress and the corresponding
friction coefficients are given by

(15) |

(16) |

The transport equation for temperature (and salinity, ) is

(17) |

Turbulence Closure

The evolution of
(twice the turbulent kinetic energy density) is given by

(19) |

There are many choices of algebraic mixing length, , see for example
Xing and Davies (1996), and we have not evaluated all these options, but
instead use the Bakhmetev scale
since Proctor and James (1996) and Holt and James (1999a) show this
gives reasonable results in the North Sea.
We modify the length scale in deep water so that
;
where :

(20) |

(21) |

This maintains the surface and near bed mixing length profiles at a water depth of , (=150m for this work) into deeper water, with intermediate depths in the water column taking the maximum value at . Defining the profile in this way simulates, in an arbitrary fashion, the eddy scale becoming independent of water depth in deep water and prevents increasing without limit with . The eddy size in stratified water is limited according to the Ozmidov length scale (for dissipation, ), . Hence, we impose a limit on proportional to this, following Galperin et al. (1988):

The near bed boundary condition arises from a steady state balance
between shear and dissipation:

(23) |

(24) |

This system is closed by

(26) |