Rossby Waves
fo 2 8t/12 = O.
Y2H3 8t
107
(3.3.6)
For steady-flow problems this is trivially satisfied and the 2! layer model is
consistent. For time-dependent problems this approximation requires that
either the thickness of the third layer becomes very great, or that the density
jump across the interface between layers 2 and 3 becomes very large so that this
remaining term is negligible.
If the density jump across the interface z = z2 is very much larger than the
density jump across the interface z = z3, the two models (3.3.2) and (3.3.3)
coincide for a flat bottom ocean.
It is important to note that when the motion is time dependent, the fluid in
layer 3 is generally put into motion by the motion of the interface separating it
from layer 2 so that the model of a resting abyss is valid only for steady motion
(in the absence of dissipative coupling). This has important consequences for
the interpretation of the Rossby wave modes in the system. These waves are
important messengers of information, and they must propagate, in general,
within the circulation. The propagation of the waves is affected by the
circulation, and the properties of the propagation in turn dynamically shape
the circulation itself. The properties of the Rossby wave dynamics in the
presence of the gyre flow prove to be vital in understanding the structure of the
resulting steady circulation. The difference between steady wave motion in the
presence of the circulation and simple propagating waves in the absence of a
background steady flow is significant. There is more than a Galilean
transformation involved in this difference, for the steady motion is usually
vertically sheared, which means that the interfaces between the layers are
sloping. This variation of the layer thicknesses alters the potential vorticity
gradient which supports the Rossby waves and changes their propagation
characteristics.
3.4 Rossby Waves
Although studying problems of the steady circulation of the wind driven gyres,
some of the principal results can be understood in a particularly helpful way in
terms of the wave properties of the system. Anderson and Gill (1975) studied
the oceanic spin-up problem and its relation to the Ross by wave modes of the
system to demonstrate the way in which the Sverdrup interior is established
and comes into a steady state after an initially impulsive imposition of the wind
stress. We do not repeat their analysis here, to which the reader is referred for a
detailed discussion, but the broad outlines of their results may be understood
from an examination of the linearized quasi-geostrophic equations.
Consider the three-layer model for a flat-bottom ocean in which
dissipation due to frictional and cross-interface motion are ignored. Suppose
fo 2 8t/12 = O.
Y2H3 8t
107
(3.3.6)
For steady-flow problems this is trivially satisfied and the 2! layer model is
consistent. For time-dependent problems this approximation requires that
either the thickness of the third layer becomes very great, or that the density
jump across the interface between layers 2 and 3 becomes very large so that this
remaining term is negligible.
If the density jump across the interface z = z2 is very much larger than the
density jump across the interface z = z3, the two models (3.3.2) and (3.3.3)
coincide for a flat bottom ocean.
It is important to note that when the motion is time dependent, the fluid in
layer 3 is generally put into motion by the motion of the interface separating it
from layer 2 so that the model of a resting abyss is valid only for steady motion
(in the absence of dissipative coupling). This has important consequences for
the interpretation of the Rossby wave modes in the system. These waves are
important messengers of information, and they must propagate, in general,
within the circulation. The propagation of the waves is affected by the
circulation, and the properties of the propagation in turn dynamically shape
the circulation itself. The properties of the Rossby wave dynamics in the
presence of the gyre flow prove to be vital in understanding the structure of the
resulting steady circulation. The difference between steady wave motion in the
presence of the circulation and simple propagating waves in the absence of a
background steady flow is significant. There is more than a Galilean
transformation involved in this difference, for the steady motion is usually
vertically sheared, which means that the interfaces between the layers are
sloping. This variation of the layer thicknesses alters the potential vorticity
gradient which supports the Rossby waves and changes their propagation
characteristics.
3.4 Rossby Waves
Although studying problems of the steady circulation of the wind driven gyres,
some of the principal results can be understood in a particularly helpful way in
terms of the wave properties of the system. Anderson and Gill (1975) studied
the oceanic spin-up problem and its relation to the Ross by wave modes of the
system to demonstrate the way in which the Sverdrup interior is established
and comes into a steady state after an initially impulsive imposition of the wind
stress. We do not repeat their analysis here, to which the reader is referred for a
detailed discussion, but the broad outlines of their results may be understood
from an examination of the linearized quasi-geostrophic equations.
Consider the three-layer model for a flat-bottom ocean in which
dissipation due to frictional and cross-interface motion are ignored. Suppose