132
ERIC BLAYO AND LAURENT DEBREU
ν 0 /
√
2 in Ω loc , except in two small transition zones of width δ, where
it varies smoothly between ν 0 and ν 0 /
√
2. This problem has a unique
solution (Brezis, 1983), denoted u ref , which is plotted on Figure 2b.
The elliptic nature of this problem amplifies the influence of the boundary conditions, which will help highlighting the differences between the
nesting approaches described in §2.
Open boundary problem.
Solve −ν(x)u
obc (x)+u obc (x) = sin nπx,
x ∈]a, b[, with OBCs at a and b. Such OBCs can be for example
Dirichlet conditions u obc (a) = α 0 , u obc (b) = β 0 , or Neumann conditions
u
obc (a) = α 1 , u
obc (b) = β 1 . If the external data are perfect ( [α 0 , β 0 ] =
[u ref (a), u ref (b)] or [α 1 , β 1 ] = [u
ref (a), u
ref (b)] ) then we get the true solution u ref . We have plotted in Figure 2b the case of imperfect Dirichlet
data α 0 = β 0 = 0.
One-way / two-way nesting.
Since the problem is not time dependent, both one-way and two-way approaches yield the same solution
u nes , defined by :
−ν 0 u
ext (x) + u ext (x) = sin nπx, x ∈]0, 1[
u ext (0) = u ext (1) = 0
−ν(x)u
nes (x) + u nes (x) = sin nπx, x ∈]a, b[
B a u nes (a) = B a u ext (a) and B b u nes (b) = B b u ext (b)
(10)
We have plotted in Figure 2b the cases B a = B b = Id and B a = B b =
∂/∂n. As can be seen clearly, these methods, which are all supposed to
approximate the true problem (9), yield quite different solutions, which
can differ from u ref both in Ω loc and Ω ext . Note also that the true
problem (9), reformulated as (5), requires two BCs at a and b, while the
approximate formulations require only one BC.
The same type of comparison is displayed in Figure 3, but for the
realistic testcase of a high resolution model of the bay of Biscay coupled
with an eddy-permitting model of the North Atlantic.
3.
The open boundary problem
Let us now focus on the main point, central in all approaches, namely
the choice of the open boundary operators B in (6)-(7)-(8). This is a
difficult problem, which has been the subject of numerous studies for
more than 30 years, ranging from purely mathematical approaches to
specific modelling applications. Mathematical results are often obtained
for simplified equations (e.g. linearized and/or inviscid). They generally
address the derivation of OBCs, and the well-posedness of the model
equations using these OBCs. Note that the well-posedness of the system
ERIC BLAYO AND LAURENT DEBREU
ν 0 /
√
2 in Ω loc , except in two small transition zones of width δ, where
it varies smoothly between ν 0 and ν 0 /
√
2. This problem has a unique
solution (Brezis, 1983), denoted u ref , which is plotted on Figure 2b.
The elliptic nature of this problem amplifies the influence of the boundary conditions, which will help highlighting the differences between the
nesting approaches described in §2.
Open boundary problem.
Solve −ν(x)u
obc (x)+u obc (x) = sin nπx,
x ∈]a, b[, with OBCs at a and b. Such OBCs can be for example
Dirichlet conditions u obc (a) = α 0 , u obc (b) = β 0 , or Neumann conditions
u
obc (a) = α 1 , u
obc (b) = β 1 . If the external data are perfect ( [α 0 , β 0 ] =
[u ref (a), u ref (b)] or [α 1 , β 1 ] = [u
ref (a), u
ref (b)] ) then we get the true solution u ref . We have plotted in Figure 2b the case of imperfect Dirichlet
data α 0 = β 0 = 0.
One-way / two-way nesting.
Since the problem is not time dependent, both one-way and two-way approaches yield the same solution
u nes , defined by :
−ν 0 u
ext (x) + u ext (x) = sin nπx, x ∈]0, 1[
u ext (0) = u ext (1) = 0
−ν(x)u
nes (x) + u nes (x) = sin nπx, x ∈]a, b[
B a u nes (a) = B a u ext (a) and B b u nes (b) = B b u ext (b)
(10)
We have plotted in Figure 2b the cases B a = B b = Id and B a = B b =
∂/∂n. As can be seen clearly, these methods, which are all supposed to
approximate the true problem (9), yield quite different solutions, which
can differ from u ref both in Ω loc and Ω ext . Note also that the true
problem (9), reformulated as (5), requires two BCs at a and b, while the
approximate formulations require only one BC.
The same type of comparison is displayed in Figure 3, but for the
realistic testcase of a high resolution model of the bay of Biscay coupled
with an eddy-permitting model of the North Atlantic.
3.
The open boundary problem
Let us now focus on the main point, central in all approaches, namely
the choice of the open boundary operators B in (6)-(7)-(8). This is a
difficult problem, which has been the subject of numerous studies for
more than 30 years, ranging from purely mathematical approaches to
specific modelling applications. Mathematical results are often obtained
for simplified equations (e.g. linearized and/or inviscid). They generally
address the derivation of OBCs, and the well-posedness of the model
equations using these OBCs. Note that the well-posedness of the system