79
Equations (AI)-(A2) are depth-independent, and are mainly concerned with motions
associated with the slope of the ocean surface. This is the reason that the ocean phenomena
obeying these equations are referred to as the "external mode", as opposed to the "internal
mode", in which the slope of density surfaces is the main source of motion.
To grasp the dynamics of the external mode, as well as that of the internal mode, it is helpful
to examine wavelike phenomena. It may be seen that the internal waves, i.e., the waves of the
internal mode, propagate at a phase speed that does not exceed a few meters per second. Note
in passing that this velocity is of the same order of magnitude as that of the fastest advective
processes. On the other hand, the external mode can sustain waves propagating as fast as
several hundreds of meters per second. Now, most numerical stability criteria are of the form
!:u
M ~ c'
p
(A4)
where c is the fastets propagation speed of the phenomena considered. Therefore, the time step
of the n~erical algorithm will be constrained by the external mode processes.
If the fast external waves could be ignored, the allowable time step would be 10 - 100 times
larger. This situation is so frustrating that several methods for circumventing the external wave
constraint, and hence for speeding up ocean models, have been suggested. However, before
outlining some of them, it is worth investigating the wave phenomena associated with the
external mode.
Equations (Al)-(A2) may be simplified and linearized to
a1] au av
0,
(AS)
x- +
+ -
=
at ax ay
au _ fV
h a1]
(A6)
at
-g ax'
av + fU =
h a1]
(A7)
at
-g ay'
where h, U and V denote the ocean depth - assumed constant - and the two components of
the transport, respectively. Of course, (AS)-(A7) are formally equivalent to (8)-(10), except
that, in the latter, the Coriolis factor is considered as constant - as it should be in the f-plane
approximation. In (AS), the coefficient X is normally equal to 1, but may be set to 0 when the
"rigid lid" approximation is introduced, as will be done below.
Appropriate manipulations of (AS)-(A 7) lead to (Longuet-Higgins, 1965)
...L (~ + f) av = V2 av + fJ av
g h a?
at
at
ax
(A8)
Since f3y «fo' we may consider thatf = fo in the equation above. The latter then has constant
coefficient, so that plane wave solutions may be sought. For this solution, let ro, k and k
represent the angular frequency, and the wavenumbers in both space directions. The di~persio~
relation of the waves under study reads
a __ 2 2
gJi (ur - fo ) ro
(k 2 + k 2) ro + f3 k .
x
y
x
(A9)
Equations (AI)-(A2) are depth-independent, and are mainly concerned with motions
associated with the slope of the ocean surface. This is the reason that the ocean phenomena
obeying these equations are referred to as the "external mode", as opposed to the "internal
mode", in which the slope of density surfaces is the main source of motion.
To grasp the dynamics of the external mode, as well as that of the internal mode, it is helpful
to examine wavelike phenomena. It may be seen that the internal waves, i.e., the waves of the
internal mode, propagate at a phase speed that does not exceed a few meters per second. Note
in passing that this velocity is of the same order of magnitude as that of the fastest advective
processes. On the other hand, the external mode can sustain waves propagating as fast as
several hundreds of meters per second. Now, most numerical stability criteria are of the form
!:u
M ~ c'
p
(A4)
where c is the fastets propagation speed of the phenomena considered. Therefore, the time step
of the n~erical algorithm will be constrained by the external mode processes.
If the fast external waves could be ignored, the allowable time step would be 10 - 100 times
larger. This situation is so frustrating that several methods for circumventing the external wave
constraint, and hence for speeding up ocean models, have been suggested. However, before
outlining some of them, it is worth investigating the wave phenomena associated with the
external mode.
Equations (Al)-(A2) may be simplified and linearized to
a1] au av
0,
(AS)
x- +
+ -
=
at ax ay
au _ fV
h a1]
(A6)
at
-g ax'
av + fU =
h a1]
(A7)
at
-g ay'
where h, U and V denote the ocean depth - assumed constant - and the two components of
the transport, respectively. Of course, (AS)-(A7) are formally equivalent to (8)-(10), except
that, in the latter, the Coriolis factor is considered as constant - as it should be in the f-plane
approximation. In (AS), the coefficient X is normally equal to 1, but may be set to 0 when the
"rigid lid" approximation is introduced, as will be done below.
Appropriate manipulations of (AS)-(A 7) lead to (Longuet-Higgins, 1965)
...L (~ + f) av = V2 av + fJ av
g h a?
at
at
ax
(A8)
Since f3y «fo' we may consider thatf = fo in the equation above. The latter then has constant
coefficient, so that plane wave solutions may be sought. For this solution, let ro, k and k
represent the angular frequency, and the wavenumbers in both space directions. The di~persio~
relation of the waves under study reads
a __ 2 2
gJi (ur - fo ) ro
(k 2 + k 2) ro + f3 k .
x
y
x
(A9)