Strategy for Regional Seasonal Forecasts
183
formulae using a sufficient number of quadrature points. On the other hand, for the
relation by using the so-called Gaussian grid points.
iii) Semi-Lagrangian method
More recently a new approach was developed by Robert (1982), Bates and
McDonald (1982), Bates (1984) and Robert et al. (1985), based on the semiLagrangian method. This method permits treating advective terms without any
limit; there is no computational instability, i.e., no C-F-L. (Courant-FriedrichLevy) constraint. Therefore, gre ater computational efficiency can be obtained by
using larger time step.
Using this technique a multi-level primitive equation model can be run with a
time steps 25 times larger than those allowed by the C-F-L criterion in an explicit
scheme. Combining it with the semi-implicit algorithm for the gravity waves, the
semi-Lagrangian method is very effective.
In a Lagrangian scheme, a displacement of a particle, a, b and c in three dimensional space is determined by the advection velocity, (u, V,ffi), where ffi == dp . The
dt
displacement is calculated by an iterative process. Upstream values of u, v, ffi and
the dependent variable Q are estimated by interpolation, using an x-y-O' grid. In this
sense, this method is called "semi-Lagrangian", as opposed to real "Lagrangian"
referring to a-b-c coordinate.
Advantages (i) Moisture and tracer transports are positive definite (ii) The same
advection calculation is applied three-dimensionally, as opposed to two-dimensionally in the "spectral model", (iii) Substantial reduction of gridpoints near the
poles, because of no constraints due to Gaussan grids or alias free calculation, and
(iv) Calculation is economical due to no computational instability.
Disadvantages: (i) No guarantee of conservation of kinetic energy, though in
practice, good conservation is obtained with high spatial resolution, and (ii) Substantial amount of overhead for the preparation.
10.2.4 Surface boundary conditions of oceanic GeM (z-coordinate)
The model was developed by Bryan (1969), subsequently, by Semtner (1974)
and Cox (1984). Pacanowski et al. (1993) have recent1y generalized the computer
codes, MOM (Modular Ocean Model), which is being extensively used throughout
the modeling comunity.
a. Free or Rigid surface condition
i) Rigid lid version
No vertical motion is allowed at the top. This boundary condition automatically
eliminates high frequency motions, allowing a larger time step (in the explicit
scheme).
The velocity is divided into two components, i.e., the barotropic and the baroclinic components, i.e.,
183
formulae using a sufficient number of quadrature points. On the other hand, for the
relation by using the so-called Gaussian grid points.
iii) Semi-Lagrangian method
More recently a new approach was developed by Robert (1982), Bates and
McDonald (1982), Bates (1984) and Robert et al. (1985), based on the semiLagrangian method. This method permits treating advective terms without any
limit; there is no computational instability, i.e., no C-F-L. (Courant-FriedrichLevy) constraint. Therefore, gre ater computational efficiency can be obtained by
using larger time step.
Using this technique a multi-level primitive equation model can be run with a
time steps 25 times larger than those allowed by the C-F-L criterion in an explicit
scheme. Combining it with the semi-implicit algorithm for the gravity waves, the
semi-Lagrangian method is very effective.
In a Lagrangian scheme, a displacement of a particle, a, b and c in three dimensional space is determined by the advection velocity, (u, V,ffi), where ffi == dp . The
dt
displacement is calculated by an iterative process. Upstream values of u, v, ffi and
the dependent variable Q are estimated by interpolation, using an x-y-O' grid. In this
sense, this method is called "semi-Lagrangian", as opposed to real "Lagrangian"
referring to a-b-c coordinate.
Advantages (i) Moisture and tracer transports are positive definite (ii) The same
advection calculation is applied three-dimensionally, as opposed to two-dimensionally in the "spectral model", (iii) Substantial reduction of gridpoints near the
poles, because of no constraints due to Gaussan grids or alias free calculation, and
(iv) Calculation is economical due to no computational instability.
Disadvantages: (i) No guarantee of conservation of kinetic energy, though in
practice, good conservation is obtained with high spatial resolution, and (ii) Substantial amount of overhead for the preparation.
10.2.4 Surface boundary conditions of oceanic GeM (z-coordinate)
The model was developed by Bryan (1969), subsequently, by Semtner (1974)
and Cox (1984). Pacanowski et al. (1993) have recent1y generalized the computer
codes, MOM (Modular Ocean Model), which is being extensively used throughout
the modeling comunity.
a. Free or Rigid surface condition
i) Rigid lid version
No vertical motion is allowed at the top. This boundary condition automatically
eliminates high frequency motions, allowing a larger time step (in the explicit
scheme).
The velocity is divided into two components, i.e., the barotropic and the baroclinic components, i.e.,