NESTING OCEAN MODELS
135
which become
∂φ
∂t
+ F (φ) + K(φ − φ
ext ) = 0
(13)
where K is a positive function, null on Ω loc and increasing away from
Γ (K depends on α and on the time-discretization scheme). Relaxation
methods are often performed jointly with a sponge layer approach, which
means that the model viscosity is artificially increased in Ω s , in order to
damp the local turbulent activity. Relaxation generally appears to be
one of the best methods in comparative numerical studies (e.g. R¨ oed
and Cooper, 1987; Palma and Matano, 1998; Nycander and D¨ o¨ os, 2003).
Two drawbacks of these methods must however be emphasized. The
first one is the increase of the computational cost induced by the additional layers Ω s . The ratio of this additional cost to the cost of the
initial model is roughly equal to |Ω s |/|Ω loc |, and can either be negligible
or reach some tens of percents, depending on the configuration. The
second drawback is the empirical aspect of the governing equation (13)
in the sponge layer.
Finally, note also that perfectly matched layer (PML) methods, which
have been proposed quite recently in the context of electromagnetism
(Berenger, 1994), can be seen as an improvement of relaxation methods. This methodology consists basically in a convenient splitting of the
equations with addition of relaxation terms with well-chosen coefficients.
PML approach has been applied to the Euler equations (Hu, 1996, 2001)
and to the shallow water equations (Darblade et al., 1997; Navon et al.,
2004), and leads to improved results in academic test cases. It must now
be validated in realistic configurations to get a better evaluation of its
actual effectiveness.
Radiation methods.
A very popular class of OBCs are radiation
methods. They are based on the Sommerfeld condition :
∂φ
∂t
+ c
∂φ
∂n
= 0
(14)
which corresponds to the transport of φ through Γ (n is the outward
normal vector) with the velocity c.
Orlanski (1976) proposed a numerical implementation of this condition for complex flows, including an adaptive evaluation of c. A number of variants were then derived, using alternative computations of
c, and/or taking into account the tangential derivative, and/or including an additional relaxation term (e.g. Camerlengo and O’Brien, 1980;
Miller and Thorpe, 1981; Raymond and Kuo, 1984; Barnier et al., 1998;
Marchesiello et al., 2001).
135
which become
∂φ
∂t
+ F (φ) + K(φ − φ
ext ) = 0
(13)
where K is a positive function, null on Ω loc and increasing away from
Γ (K depends on α and on the time-discretization scheme). Relaxation
methods are often performed jointly with a sponge layer approach, which
means that the model viscosity is artificially increased in Ω s , in order to
damp the local turbulent activity. Relaxation generally appears to be
one of the best methods in comparative numerical studies (e.g. R¨ oed
and Cooper, 1987; Palma and Matano, 1998; Nycander and D¨ o¨ os, 2003).
Two drawbacks of these methods must however be emphasized. The
first one is the increase of the computational cost induced by the additional layers Ω s . The ratio of this additional cost to the cost of the
initial model is roughly equal to |Ω s |/|Ω loc |, and can either be negligible
or reach some tens of percents, depending on the configuration. The
second drawback is the empirical aspect of the governing equation (13)
in the sponge layer.
Finally, note also that perfectly matched layer (PML) methods, which
have been proposed quite recently in the context of electromagnetism
(Berenger, 1994), can be seen as an improvement of relaxation methods. This methodology consists basically in a convenient splitting of the
equations with addition of relaxation terms with well-chosen coefficients.
PML approach has been applied to the Euler equations (Hu, 1996, 2001)
and to the shallow water equations (Darblade et al., 1997; Navon et al.,
2004), and leads to improved results in academic test cases. It must now
be validated in realistic configurations to get a better evaluation of its
actual effectiveness.
Radiation methods.
A very popular class of OBCs are radiation
methods. They are based on the Sommerfeld condition :
∂φ
∂t
+ c
∂φ
∂n
= 0
(14)
which corresponds to the transport of φ through Γ (n is the outward
normal vector) with the velocity c.
Orlanski (1976) proposed a numerical implementation of this condition for complex flows, including an adaptive evaluation of c. A number of variants were then derived, using alternative computations of
c, and/or taking into account the tangential derivative, and/or including an additional relaxation term (e.g. Camerlengo and O’Brien, 1980;
Miller and Thorpe, 1981; Raymond and Kuo, 1984; Barnier et al., 1998;
Marchesiello et al., 2001).