Rossby Waves
111
The second possibility (3.4.9b) represents a steady wave in which the
interface between the upper two layers is deformed by the motion and in which
the motion of the two layers is purely baroclinic, i.e., in which H, cp 1 = -Hzcp2 .
For waves with scales of the order of the gyre, i.e., with wavelengths, A., of
the order 2000 km, the speed of the zonal flow required to arrest the
"barotropic" mode is of the order of:
f3A.2
Ubarot. = 4 1!: 2 = 0(200cmjs)
(3.4.10)
for f3 = 2 x to- 13 cm-'s- 1 . This is far greater than any realistic gyre scale
velocity, and we conclude that such quasi-barotropic waves are not arrested by
the mean flow. In fact, the barotropic mode swiftly traverses a typical ocean
basin in a matter of weeks. For a basin of width 4000 km it would take only 3
weeks to cross the basin. In that time the steady barotropic mode carrying the
average Sverdrup transport is established. On the scale of the slow advective
motion of the gyre, of the order 1-10 cmjs, this is essentially an instantaneous
response.
On the other hand, for the same scales, for which A.» Ld, the required
speed for the second, baroclinic mode is given by (3.4.9b), which in this limit,
and using the definition of Ld given by (3.4.6), is:
2
f3 y 1 HtHz
Ubaroc. = f3Ld = fo2 Ht +Hz.
(3.4.11)
Characteristic values of Ld are of the order of 50 km, so that the required
arresting velocity is of the order of 5 cmjs. This is a speed much more
characteristic of the gyre scale mean flow, and it is more plausible that this
criterion could be met.
Suppose that the condition is met in the gyre: what would be its
implications? Consider a latitude line along which the Rossby wave propagates
westward. If we consider the mean gyre flow as a zonal flow, we can apply the
above results directly. Suppose the Ekman pumping is a function only of
latitude. Then on the latitude line where the Ekman pumping vanishes,
between the subpolar and subtropical gyres, the meridional velocity is zero,
and the flow is in fact strictly zonal there. This zonal velocity is of course zero
on the eastern boundary and increases in magnitude further west. Now let the
baroclinic Ross by wave start its journey westward from the eastern boundary.
Near the eastern boundary its speed exceeds that of the zonal flow, and the
Rossby waves propagate westward, turning off the lower layer flow. At some
more western longitude, x = Xn the zonal speed, which increases with distance
from the eastern boundary, may become large enough to satisfy (3.4.11). In this
case the further westward progress of the Rossby wave is arrested, and the
information that it carries about the presence of the eastern boundary is not
delivered to fluid further west than the longitude x7 • Fluid west of this point is
then sheltered from the Rossby wave that can make no further headway
111
The second possibility (3.4.9b) represents a steady wave in which the
interface between the upper two layers is deformed by the motion and in which
the motion of the two layers is purely baroclinic, i.e., in which H, cp 1 = -Hzcp2 .
For waves with scales of the order of the gyre, i.e., with wavelengths, A., of
the order 2000 km, the speed of the zonal flow required to arrest the
"barotropic" mode is of the order of:
f3A.2
Ubarot. = 4 1!: 2 = 0(200cmjs)
(3.4.10)
for f3 = 2 x to- 13 cm-'s- 1 . This is far greater than any realistic gyre scale
velocity, and we conclude that such quasi-barotropic waves are not arrested by
the mean flow. In fact, the barotropic mode swiftly traverses a typical ocean
basin in a matter of weeks. For a basin of width 4000 km it would take only 3
weeks to cross the basin. In that time the steady barotropic mode carrying the
average Sverdrup transport is established. On the scale of the slow advective
motion of the gyre, of the order 1-10 cmjs, this is essentially an instantaneous
response.
On the other hand, for the same scales, for which A.» Ld, the required
speed for the second, baroclinic mode is given by (3.4.9b), which in this limit,
and using the definition of Ld given by (3.4.6), is:
2
f3 y 1 HtHz
Ubaroc. = f3Ld = fo2 Ht +Hz.
(3.4.11)
Characteristic values of Ld are of the order of 50 km, so that the required
arresting velocity is of the order of 5 cmjs. This is a speed much more
characteristic of the gyre scale mean flow, and it is more plausible that this
criterion could be met.
Suppose that the condition is met in the gyre: what would be its
implications? Consider a latitude line along which the Rossby wave propagates
westward. If we consider the mean gyre flow as a zonal flow, we can apply the
above results directly. Suppose the Ekman pumping is a function only of
latitude. Then on the latitude line where the Ekman pumping vanishes,
between the subpolar and subtropical gyres, the meridional velocity is zero,
and the flow is in fact strictly zonal there. This zonal velocity is of course zero
on the eastern boundary and increases in magnitude further west. Now let the
baroclinic Ross by wave start its journey westward from the eastern boundary.
Near the eastern boundary its speed exceeds that of the zonal flow, and the
Rossby waves propagate westward, turning off the lower layer flow. At some
more western longitude, x = Xn the zonal speed, which increases with distance
from the eastern boundary, may become large enough to satisfy (3.4.11). In this
case the further westward progress of the Rossby wave is arrested, and the
information that it carries about the presence of the eastern boundary is not
delivered to fluid further west than the longitude x7 • Fluid west of this point is
then sheltered from the Rossby wave that can make no further headway