118
3. Beyond the One-Way WaveEquation
equation
81/1 + U 81/1 + V 81/1 = 0,
8t
8x
8y
(3.20)
which may be approximated by leapfrog-tirne, centered second-order space differencing as
02tcP + U02xcP + V 02ycP = O.
(3.21)
Let JL = U ßt/ tu and v = V l!.t/ l!.y be the Courant numbers for flow parallel to
the x- and y-axes. The finite-difference equation (3.21) has discrcte solutions of
the form
,j,j
_ ei(kmßxHnßy-wjM)
o/m ,n -
,
provided that w, k, and l satisfy the discrete dispersion relation
sin(wl!.t) = JL sin(kßx) + v sin(ll!.y).
A necessary condition for stability is that w be real, or equivalently, that
(3.22)
IJLI + lvl 1.
(3.23)
As discussed in connection with (2.92), the sufficient condition for stability actually requires strict inequality in (3.23) in order to avoid weakly growing modes
such as
= j cos []f(rn + n - j)/2] ,
which is a solution to (3.21) when JL = v = !.The distinction between strict
inequality and the condition given in (3.23) is, however, of little practical significance.
In order to better compare this stability condition with that for one-dimensional
advection, suppose that l!.x = l!.y = l!.s and express the wind components in
tenns of wind speed c and direction () such that U = c cos () and V = c sin ().
Then the stability condition may be written
c(i cos e] + Isin()l) < 1.
The left side of the preceding inequality is maximized when the wind blows diagonally across the mesh. If C denotes abound on the magnitude of the twodimensional velocity vector, the stability condition becomes eßt/ Ss < 1/../2.
Comparing this with the corresponding result for one-dimensional flow, it is apparent that the maximum stabletime step in the two-dimensional case is decreased
by a factor of 1/../2.
The stability condition for two-dimensional flow is more restrictive than that for
the one-dimensional case because shorter-wavelength disturbances are present on
the two-dimensional mesh. The manner in which two-dimensional grids can support wavelengths shorter than 2l!.x is illustrated in Fig. 3.3. In the case shown in
Fig. 3.3, ßx = l!.y = Ss , Grid points beneath a wave crest are indicated by solid
3. Beyond the One-Way WaveEquation
equation
81/1 + U 81/1 + V 81/1 = 0,
8t
8x
8y
(3.20)
which may be approximated by leapfrog-tirne, centered second-order space differencing as
02tcP + U02xcP + V 02ycP = O.
(3.21)
Let JL = U ßt/ tu and v = V l!.t/ l!.y be the Courant numbers for flow parallel to
the x- and y-axes. The finite-difference equation (3.21) has discrcte solutions of
the form
,j,j
_ ei(kmßxHnßy-wjM)
o/m ,n -
,
provided that w, k, and l satisfy the discrete dispersion relation
sin(wl!.t) = JL sin(kßx) + v sin(ll!.y).
A necessary condition for stability is that w be real, or equivalently, that
(3.22)
IJLI + lvl 1.
(3.23)
As discussed in connection with (2.92), the sufficient condition for stability actually requires strict inequality in (3.23) in order to avoid weakly growing modes
such as
= j cos []f(rn + n - j)/2] ,
which is a solution to (3.21) when JL = v = !.The distinction between strict
inequality and the condition given in (3.23) is, however, of little practical significance.
In order to better compare this stability condition with that for one-dimensional
advection, suppose that l!.x = l!.y = l!.s and express the wind components in
tenns of wind speed c and direction () such that U = c cos () and V = c sin ().
Then the stability condition may be written
c(i cos e] + Isin()l) < 1.
The left side of the preceding inequality is maximized when the wind blows diagonally across the mesh. If C denotes abound on the magnitude of the twodimensional velocity vector, the stability condition becomes eßt/ Ss < 1/../2.
Comparing this with the corresponding result for one-dimensional flow, it is apparent that the maximum stabletime step in the two-dimensional case is decreased
by a factor of 1/../2.
The stability condition for two-dimensional flow is more restrictive than that for
the one-dimensional case because shorter-wavelength disturbances are present on
the two-dimensional mesh. The manner in which two-dimensional grids can support wavelengths shorter than 2l!.x is illustrated in Fig. 3.3. In the case shown in
Fig. 3.3, ßx = l!.y = Ss , Grid points beneath a wave crest are indicated by solid