NESTING OCEAN MODELS
139
characteristic variables will be either inflowing or outflowing, depending
on the sign of λ k .
A fundamental point is that, for a hyperbolic open boundary problem
to be well-posed, one must prescribe as many boundary conditions as the
number of incoming characteristics. This result is in fact quite intuitive:
the solution can be decomposed into outgoing and incoming characteristics; information on the former is available within the computation
domain, and no additional condition is required, while information on
the latter is not available, and mustbe specified.
Consistency with external data.
The second keypoint concerns
the connection with external data. It appears that a reasonable choice
consists in imposing the consistency locally all along the boundary. This
means that the OBC is of the form
Bφ = Bφ
ext
(23)
where B is the open boundary operator. B = Id corresponds to the
continuity of φ through the boundary, and B = ∂/∂n to the continuity
of the flux. Such a formulation (23) is quite natural for example if we
consider that the external data φ
ext represents some steady state or far
field solution φ ∞ . In that case, as detailed for example by Engquist and
Halpern (1988), if we want the model solution to converge to the steady
state solution as t → ∞, then the OBC must also be satisfied by φ ∞ .
Used together with the point of view of characteristic variables presented previously, this condition (23) leads to recommending OBCs of
the form
Bw = Bw
ext
(24)
where w is any incoming characteristic variable of the governing equations.
The extension to non-hyperbolic systems, like for example the NavierStokes equations, is not trivial. A logical approximation consists however
in considering only the hyperbolic part of thesystem, and to use the same
procedures as for the hyperbolic case.
Revisiting usual OBCs.
The preceding criteria give a new light
on usual OBCs. It appears indeed that:
the Sommerfeld condition (14) corresponds to prescribing to zero
the incoming characteristic of the wave equation. That is why it
is legitimate for wave equations but not for other systems.
139
characteristic variables will be either inflowing or outflowing, depending
on the sign of λ k .
A fundamental point is that, for a hyperbolic open boundary problem
to be well-posed, one must prescribe as many boundary conditions as the
number of incoming characteristics. This result is in fact quite intuitive:
the solution can be decomposed into outgoing and incoming characteristics; information on the former is available within the computation
domain, and no additional condition is required, while information on
the latter is not available, and mustbe specified.
Consistency with external data.
The second keypoint concerns
the connection with external data. It appears that a reasonable choice
consists in imposing the consistency locally all along the boundary. This
means that the OBC is of the form
Bφ = Bφ
ext
(23)
where B is the open boundary operator. B = Id corresponds to the
continuity of φ through the boundary, and B = ∂/∂n to the continuity
of the flux. Such a formulation (23) is quite natural for example if we
consider that the external data φ
ext represents some steady state or far
field solution φ ∞ . In that case, as detailed for example by Engquist and
Halpern (1988), if we want the model solution to converge to the steady
state solution as t → ∞, then the OBC must also be satisfied by φ ∞ .
Used together with the point of view of characteristic variables presented previously, this condition (23) leads to recommending OBCs of
the form
Bw = Bw
ext
(24)
where w is any incoming characteristic variable of the governing equations.
The extension to non-hyperbolic systems, like for example the NavierStokes equations, is not trivial. A logical approximation consists however
in considering only the hyperbolic part of thesystem, and to use the same
procedures as for the hyperbolic case.
Revisiting usual OBCs.
The preceding criteria give a new light
on usual OBCs. It appears indeed that:
the Sommerfeld condition (14) corresponds to prescribing to zero
the incoming characteristic of the wave equation. That is why it
is legitimate for wave equations but not for other systems.