426
8. Nonreflecting Boundary Conditions
This relationship between P and wis nonlocal, in the sense that it cannot be
imposed as an algebraic or differential relation involving P and w on the physical
mesh,
The one-way dispersion relat ions (8.44) and (8.54) are also nonlocal formulae
that might be useful as radiation boundary conditions if it were possible to transform the values of w(x , z. t) on the computational mesh to and from the space of
dual variables w(k, m, w). There is, however, no way to determine the frequency
dependence of the dual variables, because the grid-point data are never simultaneously available at more than one or two time levels, and as a consequence, (8.44)
and (8.54) have no direct practical utility. In contrast, the nonlocal condition (8.55)
can be easily used in practical computations, because neither w nor m appear in
that formula. All that is required to use this boundary condition in a numerical
model with periodic lateral boundaries is to compute the Fourier transform of the
w values along the top row of the computational mesh, evaluate Pusing (8.55),
and then obtain the values of P along the top row of the computational mesh from
an inverse Fourier transform. Further details about the numerical implementation
of this boundary condition are provided in Section 8.3.3 .
The boundary condition (8.55) perfectly transmits linear hydrostatic waves
through the upper boundary, but nonhydrostatic waves will be partially reflected
back into the domain. The strength of the partial reflection can be determined as
folIows. Since the vertical group velocity obtained without making the hydrostatic
approximation is
aw
Nkm
am = =f (k2 + m2)3/2 '
the gravity-wave dispersion relation can be limited to those waves with upward
group velocities by taking the negative root in (8.42) when sgn(k) =sgn(m) and
the positive root when sgn(k) = -sgn(m). In both cases, using the dispersion
relation for the wave with upward group velocity to substitute for (w - Uk)/ kin
(8.53) yields the correct radiation condition for nonhydrostatic waves,
A
N
P=-sw
Ikl '
(8.56)
where
.
Iml
S= (k2 + m 2)1/2
A similar derivation shows that the pressure and vertical velocity in downwardpropagating waves are correlated such that
A
N A
P = --sw .
(8.57)
Ikl
Now suppose that a nonhydrostatic wave ofthe form (8.41) encounters the top
boundary. In order to satisfy the hydrostatic open boundary condition, a second
downward-propagating wave with the same horizontal wave number must also be
8. Nonreflecting Boundary Conditions
This relationship between P and wis nonlocal, in the sense that it cannot be
imposed as an algebraic or differential relation involving P and w on the physical
mesh,
The one-way dispersion relat ions (8.44) and (8.54) are also nonlocal formulae
that might be useful as radiation boundary conditions if it were possible to transform the values of w(x , z. t) on the computational mesh to and from the space of
dual variables w(k, m, w). There is, however, no way to determine the frequency
dependence of the dual variables, because the grid-point data are never simultaneously available at more than one or two time levels, and as a consequence, (8.44)
and (8.54) have no direct practical utility. In contrast, the nonlocal condition (8.55)
can be easily used in practical computations, because neither w nor m appear in
that formula. All that is required to use this boundary condition in a numerical
model with periodic lateral boundaries is to compute the Fourier transform of the
w values along the top row of the computational mesh, evaluate Pusing (8.55),
and then obtain the values of P along the top row of the computational mesh from
an inverse Fourier transform. Further details about the numerical implementation
of this boundary condition are provided in Section 8.3.3 .
The boundary condition (8.55) perfectly transmits linear hydrostatic waves
through the upper boundary, but nonhydrostatic waves will be partially reflected
back into the domain. The strength of the partial reflection can be determined as
folIows. Since the vertical group velocity obtained without making the hydrostatic
approximation is
aw
Nkm
am = =f (k2 + m2)3/2 '
the gravity-wave dispersion relation can be limited to those waves with upward
group velocities by taking the negative root in (8.42) when sgn(k) =sgn(m) and
the positive root when sgn(k) = -sgn(m). In both cases, using the dispersion
relation for the wave with upward group velocity to substitute for (w - Uk)/ kin
(8.53) yields the correct radiation condition for nonhydrostatic waves,
A
N
P=-sw
Ikl '
(8.56)
where
.
Iml
S= (k2 + m 2)1/2
A similar derivation shows that the pressure and vertical velocity in downwardpropagating waves are correlated such that
A
N A
P = --sw .
(8.57)
Ikl
Now suppose that a nonhydrostatic wave ofthe form (8.41) encounters the top
boundary. In order to satisfy the hydrostatic open boundary condition, a second
downward-propagating wave with the same horizontal wave number must also be