128
ERIC BLAYO AND LAURENT DEBREU
2.
A classification of nesting problems
2.1
General framework
We are interested in representing as accurately as possible the ocean
in a local domain Ω loc . The circulation is supposed to be described on
a time period [0, T ] by a model which can be written symbolically
L loc u loc = f loc inΩ loc × [0, T ]
( 1 )
with convenient initial conditions at t = 0. L loc is a partial differential
operator, u loc is the state variable, and f loc the model forcing. The
conditions at the solid boundaries will never be mentioned in this note,
since they do not interfere with our subject.
Since Ω loc is not closed, a portion of its boundary does not correspond
to a solid wall, and has no physical reality. This artificial interface, also
called open boundary (OB), is denoted Γ. The local solution u loc is
thus in interaction with the external ocean through Γ, and the difficulty
consists in adequately representing this interaction in order to get a good
approximation of u loc in Ω loc × [0, T ].
We also assume that we have at our disposal a (probably less accurate)
representation of the external ocean, either under the form of some data
u ext or of an external model
L ext u ext = f ext in Ω ext × [0, T ]
( 2 )
where Ω ext is an external oceanic domain. Note that, in our notations,
Ω loc and Ω ext do not overlap (Figure 1).
The best way to solve the local problem is then probably to use an
inverse approach (e.g. Bennett, 2002), i.e. for example
Find u loc that minimizes L loc u loc −f loc
2
Ω loc ×[0,T ] +ε u loc −u ext
2
Γ×[0,T ] (3)
where the norms are defined conveniently and take into account some
statistical knowledge on the errors on u ext and on the model (1), and
where ε is a weighting factor. One can also consider that the model is
perfect, and minimize only u loc − u ext 2
Γ×[0,T ] , i.e. control the boundary
values, under the constraint (1) (e.g. Taillandier et al., 2004).
However solving such an inverse problem is quite difficult and expensive. That is why ocean modellers usually use direct approaches. The
goal is then to find u loc satisfying (1) that connects adequately to u ext
through Γ. The mathematical formulation of this problem is generally
not expressed clearly in actual applications. Since Γ has no physical reality, the connection between u ext and u loc should be as smooth as possible,
ERIC BLAYO AND LAURENT DEBREU
2.
A classification of nesting problems
2.1
General framework
We are interested in representing as accurately as possible the ocean
in a local domain Ω loc . The circulation is supposed to be described on
a time period [0, T ] by a model which can be written symbolically
L loc u loc = f loc inΩ loc × [0, T ]
( 1 )
with convenient initial conditions at t = 0. L loc is a partial differential
operator, u loc is the state variable, and f loc the model forcing. The
conditions at the solid boundaries will never be mentioned in this note,
since they do not interfere with our subject.
Since Ω loc is not closed, a portion of its boundary does not correspond
to a solid wall, and has no physical reality. This artificial interface, also
called open boundary (OB), is denoted Γ. The local solution u loc is
thus in interaction with the external ocean through Γ, and the difficulty
consists in adequately representing this interaction in order to get a good
approximation of u loc in Ω loc × [0, T ].
We also assume that we have at our disposal a (probably less accurate)
representation of the external ocean, either under the form of some data
u ext or of an external model
L ext u ext = f ext in Ω ext × [0, T ]
( 2 )
where Ω ext is an external oceanic domain. Note that, in our notations,
Ω loc and Ω ext do not overlap (Figure 1).
The best way to solve the local problem is then probably to use an
inverse approach (e.g. Bennett, 2002), i.e. for example
Find u loc that minimizes L loc u loc −f loc
2
Ω loc ×[0,T ] +ε u loc −u ext
2
Γ×[0,T ] (3)
where the norms are defined conveniently and take into account some
statistical knowledge on the errors on u ext and on the model (1), and
where ε is a weighting factor. One can also consider that the model is
perfect, and minimize only u loc − u ext 2
Γ×[0,T ] , i.e. control the boundary
values, under the constraint (1) (e.g. Taillandier et al., 2004).
However solving such an inverse problem is quite difficult and expensive. That is why ocean modellers usually use direct approaches. The
goal is then to find u loc satisfying (1) that connects adequately to u ext
through Γ. The mathematical formulation of this problem is generally
not expressed clearly in actual applications. Since Γ has no physical reality, the connection between u ext and u loc should be as smooth as possible,