280
PIERRE BRASSEUR
this stage the practical representation of the mathematical objects introduced in the previous section, with examples taken from high-resolution
circulation systems developed from an operational perspective.
3.1
The state vector and model operator
The state vector x is a discrete representation of the variables involved
in the description of the system state. It characterizes information about
the space variability of the physical or biological quantities, and about
the multivariate relationships between the dierent dynamical variables.
In a numerical model of the Primitive Equations (PE), x typically contains the 3D discretized temperature, salinity, zonal and meridional
velocities, and the 2D sea-surface height or barotropic streamfunction
field computed prognostically at every gridpoint of a finite-dierence
mesh. The typical size of a PE state vector n is approximately given by
N X × N Y × (N Z × 4 variables + 1 variable), where N X , N Y and N Z are
the horizontal and vertical grid dimensions. The length n of the state
vector is typically 10 6 to 10 8 in scientific or operational applications.
A biological state vector would contain, for instance, the concentration
distributions of nutrients, plankton, dissolved and particulate matter,
etc. [Carmillet et al., 2001].
Let us consider, for example, the 1/12
model configuration of the
North Atlantic ocean that has been developed using the HYbrid Coordinate Ocean Model (HYCOM) with a horizontal grid size of approximately 1400 x 1400, and 26 layers in the vertical direction. Figure 3
shows a snapshot extracted from a simulation, illustrating the space
variability described by the multivariate HYCOM state vector.
Note that the vertical hybrid coordinate used in HYCOM is a combination of geometric and dynamic vertical coordinates that evolves dynamically with the state of the system itself [Bleck, 2002]. Due to the
occurrence of outcropping layers at the base of the mixed layer, the
number of discrete variables may be dierent from one timestep to
another, and the dimension of the state vector is therefore dynamicallydependent. This feature slightly complicates the handling of the state
vector [Brankart et al., 2003; Birol et al., 2004], compared with more
conventional models based on static vertical coordinates such as OPA.
The size problems arise mainly from equations (18), (20) and (21)
which involve the manipulation of n × n matrices. With a state dimension
of 10 8 , the storage of a full n × n matrix would require a memory of
10 5 gigabytes, representing about 1000 times the capacity of the largest
computers available today. For this reason, it is necessary to by-pass the
explicit representation of such matrices in the algorithms. For instance,
PIERRE BRASSEUR
this stage the practical representation of the mathematical objects introduced in the previous section, with examples taken from high-resolution
circulation systems developed from an operational perspective.
3.1
The state vector and model operator
The state vector x is a discrete representation of the variables involved
in the description of the system state. It characterizes information about
the space variability of the physical or biological quantities, and about
the multivariate relationships between the dierent dynamical variables.
In a numerical model of the Primitive Equations (PE), x typically contains the 3D discretized temperature, salinity, zonal and meridional
velocities, and the 2D sea-surface height or barotropic streamfunction
field computed prognostically at every gridpoint of a finite-dierence
mesh. The typical size of a PE state vector n is approximately given by
N X × N Y × (N Z × 4 variables + 1 variable), where N X , N Y and N Z are
the horizontal and vertical grid dimensions. The length n of the state
vector is typically 10 6 to 10 8 in scientific or operational applications.
A biological state vector would contain, for instance, the concentration
distributions of nutrients, plankton, dissolved and particulate matter,
etc. [Carmillet et al., 2001].
Let us consider, for example, the 1/12
model configuration of the
North Atlantic ocean that has been developed using the HYbrid Coordinate Ocean Model (HYCOM) with a horizontal grid size of approximately 1400 x 1400, and 26 layers in the vertical direction. Figure 3
shows a snapshot extracted from a simulation, illustrating the space
variability described by the multivariate HYCOM state vector.
Note that the vertical hybrid coordinate used in HYCOM is a combination of geometric and dynamic vertical coordinates that evolves dynamically with the state of the system itself [Bleck, 2002]. Due to the
occurrence of outcropping layers at the base of the mixed layer, the
number of discrete variables may be dierent from one timestep to
another, and the dimension of the state vector is therefore dynamicallydependent. This feature slightly complicates the handling of the state
vector [Brankart et al., 2003; Birol et al., 2004], compared with more
conventional models based on static vertical coordinates such as OPA.
The size problems arise mainly from equations (18), (20) and (21)
which involve the manipulation of n × n matrices. With a state dimension
of 10 8 , the storage of a full n × n matrix would require a memory of
10 5 gigabytes, representing about 1000 times the capacity of the largest
computers available today. For this reason, it is necessary to by-pass the
explicit representation of such matrices in the algorithms. For instance,