NESTING OCEAN MODELS
129
i.e. generally continuous and differentiable. Therefore a correct direct
formulation of the problem can be the following :
Find u loc that satisfies
⎧
⎨
⎩
L loc u loc = f loc in Ω loc × [0, T ]
u loc = u ext and
∂u loc
∂n
=
∂u ext
∂n
on Γ × [0, T ]
under the constraint L ext u ext = f ext in Ω ext × [0, T ]
(4)
or equivalently :
Find u loc and u ext that satisfy
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
L loc u loc = f loc in Ω loc × [0, T ] and L ext u ext = f ext
in Ω ext × [0, T ]
with u loc = u ext and
∂u loc
∂n
=
∂u ext
∂n
on Γ × [0, T ]
(5)
where n denotes the normal direction. However, in actual applications,
the external model is not always available for online interaction. Moreover it is defined generally on Ω ext ∪ Ω loc (i.e. it fully overlaps the local
domain), and it would be quite expensive to modify it in order to avoid
this overlapping by implementing an open boundary on Γ. Therefore
most applications generally do not address thecorrect problem (5) itself,
but rather more or less approaching problems.
Remark: the operators L ext and L loc generally differ, both in their
continuous form (e.g. subgrid scale paramaterizations) and in their discretized form (the local numerical model often has a higher resolution
than the external model). Moreover the forcings f ext and f loc , and the
discretized bathymetries defining Ω ext and Ω loc can be rather different.
In that case the regularity conditions in (5) cannot be satisfied, and
the connection between u ext and u loc is unsmooth, which is of course
non-physical. That is why it is recommended to define the models and
forcings in order to ensure as far as possible the smoothness of the transition between the two models. This can be done for instance into a
transition zone defined in the vicinity of Γ.
2.2
The different approaches
The usual approaches can be classified as follows:
The open boundary problem.
This is the usual case where the
local model only is used. The problem writes
L loc u loc = f loc inΩ loc × [0, T ]
Bu loc = g
on Γ × [0, T ]
(6)
129
i.e. generally continuous and differentiable. Therefore a correct direct
formulation of the problem can be the following :
Find u loc that satisfies
⎧
⎨
⎩
L loc u loc = f loc in Ω loc × [0, T ]
u loc = u ext and
∂u loc
∂n
=
∂u ext
∂n
on Γ × [0, T ]
under the constraint L ext u ext = f ext in Ω ext × [0, T ]
(4)
or equivalently :
Find u loc and u ext that satisfy
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
L loc u loc = f loc in Ω loc × [0, T ] and L ext u ext = f ext
in Ω ext × [0, T ]
with u loc = u ext and
∂u loc
∂n
=
∂u ext
∂n
on Γ × [0, T ]
(5)
where n denotes the normal direction. However, in actual applications,
the external model is not always available for online interaction. Moreover it is defined generally on Ω ext ∪ Ω loc (i.e. it fully overlaps the local
domain), and it would be quite expensive to modify it in order to avoid
this overlapping by implementing an open boundary on Γ. Therefore
most applications generally do not address thecorrect problem (5) itself,
but rather more or less approaching problems.
Remark: the operators L ext and L loc generally differ, both in their
continuous form (e.g. subgrid scale paramaterizations) and in their discretized form (the local numerical model often has a higher resolution
than the external model). Moreover the forcings f ext and f loc , and the
discretized bathymetries defining Ω ext and Ω loc can be rather different.
In that case the regularity conditions in (5) cannot be satisfied, and
the connection between u ext and u loc is unsmooth, which is of course
non-physical. That is why it is recommended to define the models and
forcings in order to ensure as far as possible the smoothness of the transition between the two models. This can be done for instance into a
transition zone defined in the vicinity of Γ.
2.2
The different approaches
The usual approaches can be classified as follows:
The open boundary problem.
This is the usual case where the
local model only is used. The problem writes
L loc u loc = f loc inΩ loc × [0, T ]
Bu loc = g
on Γ × [0, T ]
(6)