5.13 Advanced Lateral Boundary Conditions
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a time-mean component driving the geostrophic inflow and a superposed timevariable component representing incident waves such as tides. The geostrophic part
can then be described along the boundary as a background distribution, gradually
blended in during the first few days of a simulation, and the time-variable part can
be added via a prognostic equation of the form:
∂q b
∂t
= q o ω sin (ωt)
(5.38)
where q b is boundary dynamic pressure, and q o and ω are amplitude and frequency
of the time-variable forcing. The reader is encouraged to test this type of forcing for
a simplified open channel configuration.
Inflow conditions for water properties such as temperature and salinity are
straight forward. These properties are simply prescribed as fixed boundary values
so that the inflow carries these via the advection scheme into the model domain.
Establishment of sharp density gradients near open boundaries can be avoided with
the use of an adjustment method, commonly called Rayleigh damping, of the form:
ψ
n+1
b
= ψ
n
b +
Δt
T
ψ o − ψ
n
b
(5.39)
where ψ b is the boundary value of either temperature or salinity, T is a prescribed
adjustment period, and ψ o is the target boundary value.
5.13.4 Outflow Conditions
The formulation of outflow conditions at open boundaries is not a trivial task given
that both steady currents and wave signals can simultaneously interfere with such
a boundary. Unwanted partial wave reflection at open boundaries is a common
problem. Different types of outflow boundary conditions are best demonstrated
with a focus on the propagation of long linear surface gravity waves in a channel of uniform depth h o . The dynamics of such waves can be approximated by the
equations:
∂u
∂t
= −g
∂η
∂ x
(5.40)
∂η
∂t
= −h o
∂u
∂ x
(5.41)
where u is velocity, t is time, g is acceleration due to gravity, η is seasurface elevation, and x is distance along the channel. Assumptions are that the wave period
is short compared with the Coriolis force (so that the latter can be ignored), that
the wave’s phase speed exceeds the flow speed by far (so that the nonlinear terms
can be ignored), that the hydrostatic balance holds (such that the resultant flow is
167
a time-mean component driving the geostrophic inflow and a superposed timevariable component representing incident waves such as tides. The geostrophic part
can then be described along the boundary as a background distribution, gradually
blended in during the first few days of a simulation, and the time-variable part can
be added via a prognostic equation of the form:
∂q b
∂t
= q o ω sin (ωt)
(5.38)
where q b is boundary dynamic pressure, and q o and ω are amplitude and frequency
of the time-variable forcing. The reader is encouraged to test this type of forcing for
a simplified open channel configuration.
Inflow conditions for water properties such as temperature and salinity are
straight forward. These properties are simply prescribed as fixed boundary values
so that the inflow carries these via the advection scheme into the model domain.
Establishment of sharp density gradients near open boundaries can be avoided with
the use of an adjustment method, commonly called Rayleigh damping, of the form:
ψ
n+1
b
= ψ
n
b +
Δt
T
ψ o − ψ
n
b
(5.39)
where ψ b is the boundary value of either temperature or salinity, T is a prescribed
adjustment period, and ψ o is the target boundary value.
5.13.4 Outflow Conditions
The formulation of outflow conditions at open boundaries is not a trivial task given
that both steady currents and wave signals can simultaneously interfere with such
a boundary. Unwanted partial wave reflection at open boundaries is a common
problem. Different types of outflow boundary conditions are best demonstrated
with a focus on the propagation of long linear surface gravity waves in a channel of uniform depth h o . The dynamics of such waves can be approximated by the
equations:
∂u
∂t
= −g
∂η
∂ x
(5.40)
∂η
∂t
= −h o
∂u
∂ x
(5.41)
where u is velocity, t is time, g is acceleration due to gravity, η is seasurface elevation, and x is distance along the channel. Assumptions are that the wave period
is short compared with the Coriolis force (so that the latter can be ignored), that
the wave’s phase speed exceeds the flow speed by far (so that the nonlinear terms
can be ignored), that the hydrostatic balance holds (such that the resultant flow is