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8. Nonreflecting Boundary Conditions
The use of Fourier transforms and the assumption of false periodicity can be
eliminated by approximating the factor of Ikl in (8.55) with an algebraic expression that converts (8.55) into a dispersion relation for a partial differential equa -
tion that can be solved on the physical mesh. Let Ik1be replaced by the rational
function
a) +a2k2
1+ a3k 2 '
(8.60)
where al, a2, and a3 are constants chosen to ensure that (8.60) is a good approximation to Ik l over the entire range of wave numbers that need to be transmitted
through the upper boundary. It is convenient to express the arbitrary constants a),
a2, and a3 in terms of the three wave numbers, ki , k2, and k3, for which (8.60)
would be exactly equal to IkI. Then
where
Numerical tests have suggested that an effective strategy for choosing appropriate
values for k) , ka. and k3 (and thereby specifying a) , az, and a3) is to let k3 be the
largest horizontal wave number likely to appear with significant amplitude in the
perturbations at the upper boundary and then choose k: = k3/3 and k) = k3/9 .
Replacing Ikl by (8.60) in (8.55) and taking an inverse Fourier transform yields
the local differential equation
(8.61)
The reftection generated by (8.61) in a vertically propagating hydrostatic gravity wave of horizontal wave number k is
Irl= k - k)
- - 11 k
- -
- k211 k
- - .
- k31
I k + k) k + k2 k + k3
This reftection coefficient has the same form as that for the three-operator Higdon
scheme, except that the wave number parallel to the boundary is k in the preceding
and is m in (8.48). The dependence of r on the horizontal wave number of an
incident hydrostatic wave is given by the solid curve in Fig. 8.6 , except that the
m 's appearing in the labels for the horizontal axis should be replaced with k 'so
Recall, however, that Higdon 's one-way wave equation is not suitable for use at
the upper boundary because the phase speed and group velocity of internal gravity
waves are generally opposite in sign. Equation (8.61) provides an alternative that
avoids instability while achieving the same degree of wave transmission through
the upper boundary.
8. Nonreflecting Boundary Conditions
The use of Fourier transforms and the assumption of false periodicity can be
eliminated by approximating the factor of Ikl in (8.55) with an algebraic expression that converts (8.55) into a dispersion relation for a partial differential equa -
tion that can be solved on the physical mesh. Let Ik1be replaced by the rational
function
a) +a2k2
1+ a3k 2 '
(8.60)
where al, a2, and a3 are constants chosen to ensure that (8.60) is a good approximation to Ik l over the entire range of wave numbers that need to be transmitted
through the upper boundary. It is convenient to express the arbitrary constants a),
a2, and a3 in terms of the three wave numbers, ki , k2, and k3, for which (8.60)
would be exactly equal to IkI. Then
where
Numerical tests have suggested that an effective strategy for choosing appropriate
values for k) , ka. and k3 (and thereby specifying a) , az, and a3) is to let k3 be the
largest horizontal wave number likely to appear with significant amplitude in the
perturbations at the upper boundary and then choose k: = k3/3 and k) = k3/9 .
Replacing Ikl by (8.60) in (8.55) and taking an inverse Fourier transform yields
the local differential equation
(8.61)
The reftection generated by (8.61) in a vertically propagating hydrostatic gravity wave of horizontal wave number k is
Irl= k - k)
- - 11 k
- -
- k211 k
- - .
- k31
I k + k) k + k2 k + k3
This reftection coefficient has the same form as that for the three-operator Higdon
scheme, except that the wave number parallel to the boundary is k in the preceding
and is m in (8.48). The dependence of r on the horizontal wave number of an
incident hydrostatic wave is given by the solid curve in Fig. 8.6 , except that the
m 's appearing in the labels for the horizontal axis should be replaced with k 'so
Recall, however, that Higdon 's one-way wave equation is not suitable for use at
the upper boundary because the phase speed and group velocity of internal gravity
waves are generally opposite in sign. Equation (8.61) provides an alternative that
avoids instability while achieving the same degree of wave transmission through
the upper boundary.